graphing activities homework answer key

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

5.1: Solve Systems of Equations by Graphing

• Last updated
• Save as PDF
• Page ID 15151

Learning Objectives

By the end of this section, you will be able to:

• Determine whether an ordered pair is a solution of a system of equations
• Solve a system of linear equations by graphing
• Determine the number of solutions of linear system
• Solve applications of systems of equations by graphing

Before you get started, take this readiness quiz.

• For the equation \(y=\frac{2}{3}x−4\) ⓐ is (6,0) a solution? ⓑ is (−3,−2) a solution? If you missed this problem, review Example 2.1.1 .
• Find the slope and y-intercept of the line 3x−y=12. If you missed this problem, review Example 4.5.7 .
• Find the x- and y-intercepts of the line 2x−3y=12. If you missed this problem, review Example 4.3.7 .

Determine Whether an Ordered Pair is a Solution of a System of Equations

In the section on Solving Linear Equations and Inequalities we learned how to solve linear equations with one variable. Remember that the solution of an equation is a value of the variable that makes a true statement when substituted into the equation. Now we will work with systems of linear equations , two or more linear equations grouped together.

Definition: SYSTEM OF LINEAR EQUATIONS

When two or more linear equations are grouped together, they form a system of linear equations.

We will focus our work here on systems of two linear equations in two unknowns. Later, you may solve larger systems of equations.

An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.

\[\begin{cases}{2 x+y=7} \\ {x-2 y=6}\end{cases}\]

A linear equation in two variables, like 2 x + y = 7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.

To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs ( x , y ) that make both equations true. These are called the solutions to a system of equations .

Definition: SolutionS OF A SYSTEM OF EQUATIONS

Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair ( x , y ).

To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.

Let’s consider the system below:

\[\begin{cases}{3x−y=7} \\ {x−2y=4}\end{cases}\]

Is the ordered pair (2,−1) a solution?

The ordered pair (2, −1) made both equations true. Therefore (2, −1) is a solution to this system.

Let’s try another ordered pair. Is the ordered pair (3, 2) a solution?

The ordered pair (3, 2) made one equation true, but it made the other equation false. Since it is not a solution to both equations, it is not a solution to this system.

Example \(\PageIndex{1}\)

Determine whether the ordered pair is a solution to the system: \(\begin{cases}{x−y=−1} \\ {2x−y=−5}\end{cases}\)

• (−2,−1)
• (−4,−3)

\(\begin{cases}{x−y=−1} \\ {2x−y=−5}\end{cases}\)

Try It \(\PageIndex{2}\)

Determine whether the ordered pair is a solution to the system: \(\begin{cases}{3x+y=0} \\ {x+2y=−5}\end{cases}\)

• (1,−3)

Try It \(\PageIndex{3}\)

Determine whether the ordered pair is a solution to the system: \(\begin{cases}{x−3y=−8} \\ {−3x−y=4}\end{cases}\)

• (2,−2)
• (−2,2)

Solve a System of Linear Equations by Graphing

In this chapter we will use three methods to solve a system of linear equations. The first method we’ll use is graphing. The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we’ll find the solution to the system.

Most linear equations in one variable have one solution, but we saw that some equations, called contradictions , have no solutions and for other equations, called identities , all numbers are solutions. Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown in Figure \(\PageIndex{1}\):

For the first example of solving a system of linear equations in this section and in the next two sections, we will solve the same system of two linear equations. But we’ll use a different method in each section. After seeing the third method, you’ll decide which method was the most convenient way to solve this system.

Example \(\PageIndex{4}\): How to Solve a System of Linear Equations by Graphing

Solve the system by graphing: \(\begin{cases}{2x+y=7} \\ {x−2y=6}\end{cases}\)

Try It \(\PageIndex{5}\)

Solve each system by graphing: \(\begin{cases}{x−3y=−3} \\ {x+y=5}\end{cases}\)

Try It \(\PageIndex{6}\)

Solve each system by graphing: \(\begin{cases}{−x+y=1} \\ {3x+2y=12}\end{cases}\)

The steps to use to solve a system of linear equations by graphing are shown below.

TO SOLVE A SYSTEM OF LINEAR EQUATIONS BY GRAPHING.

• Graph the first equation.
• Graph the second equation on the same rectangular coordinate system.
• Determine whether the lines intersect, are parallel, or are the same line.
• If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system.
• If the lines are parallel, the system has no solution.
• If the lines are the same, the system has an infinite number of solutions.

Example \(\PageIndex{7}\)

Solve the system by graphing: \(\begin{cases}{y=2x+1} \\ {y=4x−1}\end{cases}\)

Both of the equations in this system are in slope-intercept form, so we will use their slopes and y -intercepts to graph them. \(\begin{cases}{y=2x+1} \\ {y=4x−1}\end{cases}\)

Try It \(\PageIndex{8}\)

Solve the system by graphing: \(\begin{cases}{y=2x+2} \\ {y=-x−4}\end{cases}\)

(−2,−2)

Try It \(\PageIndex{9}\)

Solve the system by graphing: \(\begin{cases}{y=3x+3} \\ {y=-x+7}\end{cases}\)

Both equations in Example \(\PageIndex{7}\) were given in slope–intercept form. This made it easy for us to quickly graph the lines. In the next example, we’ll first re-write the equations into slope–intercept form.

Example \(\PageIndex{10}\)

Solve the system by graphing: \(\begin{cases}{3x+y=−1} \\ {2x+y=0}\end{cases}\)

We’ll solve both of these equations for \(y\) so that we can easily graph them using their slopes and y -intercepts. \(\begin{cases}{3x+y=−1} \\ {2x+y=0}\end{cases}\)

Try It \(\PageIndex{11}\)

Solve each system by graphing: \(\begin{cases}{−x+y=1} \\ {2x+y=10}\end{cases}\)

Try It \(\PageIndex{12}\)

Solve each system by graphing: \(\begin{cases}{ 2x+y=6} \\ {x+y=1}\end{cases}\)

(5,−4)

Usually when equations are given in standard form, the most convenient way to graph them is by using the intercepts. We’ll do this in Example \(\PageIndex{13}\).

Example \(\PageIndex{13}\)

Solve the system by graphing: \(\begin{cases}{x+y=2} \\ {x−y=4}\end{cases}\)

We will find the x - and y -intercepts of both equations and use them to graph the lines.

Try It \(\PageIndex{14}\)

Solve each system by graphing: \(\begin{cases}{x+y=6} \\ {x−y=2}\end{cases}\)

Try It \(\PageIndex{15}\)

Solve each system by graphing: \(\begin{cases}{x+y=2} \\ {x−y=8}\end{cases}\)

(5,−3)

Do you remember how to graph a linear equation with just one variable? It will be either a vertical or a horizontal line.

Example \(\PageIndex{16}\)

Solve the system by graphing: \(\begin{cases}{y=6} \\ {2x+3y=12}\end{cases}\)

Try It \(\PageIndex{17}\)

Solve each system by graphing: \(\begin{cases}{y=−1} \\ {x+3y=6}\end{cases}\)

(9,−1)

Try It \(\PageIndex{18}\)

Solve each system by graphing: \(\begin{cases}{x=4} \\ {3x−2y=24}\end{cases}\)

(4,−6)

In all the systems of linear equations so far, the lines intersected and the solution was one point. In the next two examples, we’ll look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.

Example \(\PageIndex{19}\)

Solve the system by graphing: \(\begin{cases}{y=\frac{1}{2}x−3} \\ {x−2y=4}\end{cases}\)

Try It \(\PageIndex{20}\)

Solve each system by graphing: \(\begin{cases}{y=-\frac{1}{4}x+2} \\ {x+4y=-8}\end{cases}\)

no solution

Try It \(\PageIndex{21}\)

Solve each system by graphing: \(\begin{cases}{y=3x−1} \\ {6x−2y=6}\end{cases}\)

Example \(\PageIndex{22}\)

Solve the system by graphing: \(\begin{cases}{y=2x−3} \\ {−6x+3y=−9}\end{cases}\)

Try It \(\PageIndex{23}\)

Solve each system by graphing: \(\begin{cases}{y=−3x−6} \\ {6x+2y=−12}\end{cases}\)

infinitely many solutions

Try It \(\PageIndex{24}\)

Solve each system by graphing: \(\begin{cases}{y=\frac{1}{2}x−4} \\ {2x−4y=16}\end{cases}\)

If you write the second equation in Example \(\PageIndex{22}\) in slope-intercept form, you may recognize that the equations have the same slope and same y -intercept.

When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are coincident. Coincident lines have the same slope and same y -intercept.

COINCIDENT LINES

Coincident lines have the same slope and same y -intercept.

Determine the Number of Solutions of a Linear System

There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. It will be helpful to determine this without graphing.

We have seen that two lines in the same plane must either intersect or are parallel. The systems of equations in Example \(\PageIndex{4}\) through Example \(\PageIndex{18}\) all had two intersecting lines. Each system had one solution.

A system with parallel lines, like Example \(\PageIndex{19}\), has no solution. What happened in Example \(\PageIndex{22}\)? The equations have coincident lines , and so the system had infinitely many solutions.

We’ll organize these results in Figure \(\PageIndex{2}\) below:

Parallel lines have the same slope but different y -intercepts. So, if we write both equations in a system of linear equations in slope–intercept form, we can see how many solutions there will be without graphing! Look at the system we solved in Example \(\PageIndex{19}\).

\(\begin{array} {cc} & \begin{cases}{y=\frac{1}{2}x−3} \\ {x−2y=4}\end{cases}\\ \text{The first line is in slope–intercept form.} &\text { If we solve the second equation for } y, \text { we get } \\ &x-2 y =4 \\ y = \frac{1}{2}x -3& x-2 y =-x+4 \\ &y =\frac{1}{2} x-2 \\ m=\frac{1}{2}, b=-3&m=\frac{1}{2}, b=-2 \end{array}\)

The two lines have the same slope but different y -intercepts. They are parallel lines.

Figure \(\PageIndex{3}\) shows how to determine the number of solutions of a linear system by looking at the slopes and intercepts.

Let’s take one more look at our equations in Example \(\PageIndex{19}\) that gave us parallel lines.

\[\begin{cases}{y=\frac{1}{2}x−3} \\ {x−2y=4}\end{cases}\]

When both lines were in slope-intercept form we had:

\[y=\frac{1}{2} x-3 \quad y=\frac{1}{2} x-2\]

Do you recognize that it is impossible to have a single ordered pair (x,y) that is a solution to both of those equations?

We call a system of equations like this an inconsistent system . It has no solution.

A system of equations that has at least one solution is called a consistent system .

CONSISTENT AND INCONSISTENT SYSTEMS

A consistent system of equations is a system of equations with at least one solution.

An inconsistent system of equations is a system of equations with no solution.

We also categorize the equations in a system of equations by calling the equations independent or dependent . If two equations are independent equations , they each have their own set of solutions. Intersecting lines and parallel lines are independent.

If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations , we get coincident lines.

INDEPENDENT AND DEPENDENT EQUATIONS

Two equations are independent if they have different solutions.

Two equations are dependent if all the solutions of one equation are also solutions of the other equation.

Let’s sum this up by looking at the graphs of the three types of systems. See Figure \(\PageIndex{4}\) and Figure \(\PageIndex{5}\).

Example \(\PageIndex{25}\)

Without graphing, determine the number of solutions and then classify the system of equations: \(\begin{cases}{y=3x−1} \\ {6x−2y=12}\end{cases}\)

\(\begin{array}{lrrl} \text{We will compare the slopes and intercepts} & \begin{cases}{y=3x−1} \\ {6x−2y=12}\end{cases} \\ \text{of the two lines.} \\ \text{The first equation is already in} \\ \text{slope-intercept form.} \\ & {y = 3x - 1}\\ \text{Write the second equation in} \text{slope–intercept form.} \\  && 6x-2y =&12 \\  && -2y =& -6x - 12 \\  &&\frac{-2y}{-2}=& \frac{-6x + 12}{-2}\\  &&y=&3x-6\\\\ \text{Find the slope and intercept of each line.} & y = 3x-1 & y=3x-6 \\ &m = 3 & m = 3 \\&b=-1 &b=-6 \\ \text{Since the slopes are the same and y-intercepts} \\ \text{are different, the lines are parallel.}\end{array}\)

A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.

Try It \(\PageIndex{26}\)

Without graphing, determine the number of solutions and then classify the system of equations.

\(\begin{cases}{y=−2x−4} \\ {4x+2y=9}\end{cases}\)

no solution, inconsistent, independent

Try It \(\PageIndex{27}\)

\(\begin{cases}{y=\frac{1}{3}x−5} \\ {x-3y=6}\end{cases}\)

Example \(\PageIndex{28}\)

Without graphing, determine the number of solutions and then classify the system of equations: \(\begin{cases}{2x+y=−3} \\ {x−5y=5}\end{cases}\)

\(\begin{array}{lrrlrl} \text{We will compare the slopes and intercepts} & \begin{cases}{2x+y=-3} \\ {x−5y=5}\end{cases} \\ \text{of the two lines.} \\ \text{Write the second equation in} \\ \text{slope–intercept form.} \\ &2x+y&=&-3 & x−5y&=&5\\ & y &=& -2x -3 & -5y &=&-x+5 \\ &&&&\frac{-5y}{-5} &=& \frac{-x + 5}{-5}\\ &&&&y&=&\frac{1}{5}x-1\\\\ \text{Find the slope and intercept of each line.} & y &=& -2x-3 & y&=&\frac{1}{5}x-1 \\ &m &=& -2 & m &=& \frac{1}{5} \\&b&=&-3 &b&=&-1 \\ \text{Since the slopes are  different, the lines intersect.}\end{array}\)

A system of equations whose graphs are intersect has 1 solution and is consistent and independent.

Try It \(\PageIndex{29}\)

\(\begin{cases}{3x+2y=2} \\ {2x+y=1}\end{cases}\)

one solution, consistent, independent

Try It \(\PageIndex{30}\)

\(\begin{cases}{x+4y=12} \\ {−x+y=3}\end{cases}\)

Example \(\PageIndex{31}\)

Without graphing, determine the number of solutions and then classify the system of equations. \(\begin{cases}{3x−2y=4} \\ {y=\frac{3}{2}x−2}\end{cases}\)

\(\begin{array}{lrrlrl} \text{We will compare the slopes and intercepts of the two lines.}& \begin{cases}{3x−2y} &=&{4} \\ {y}&=&{\frac{3}{2}x−2}\end{cases} \\ \text{Write the second equation in} \\ \text{slope–intercept form.} \\ &3x-2y&=&4 \\ & -2y &=& -3x +4 \\ &\frac{-2y}{-2} &=& \frac{-3x + 4}{-2}\\ &y&=&\frac{3}{2}x-2\\\\ \text{Find the slope and intercept of each line.} &y&=&\frac{3}{2}x-2\\ \text{Since the equations are the same, they have the same slope} \\ \text{and same y-intercept and so the lines are coincident.}\end{array}\)

A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.

Try It \(\PageIndex{32}\)

\(\begin{cases}{4x−5y=20} \\ {y=\frac{4}{5}x−4}\end{cases}\)

infinitely many solutions, consistent, dependent

Try It \(\PageIndex{33}\)

\(\begin{cases}{ −2x−4y=8} \\ {y=−\frac{1}{2}x−2}\end{cases}\)

Solve Applications of Systems of Equations by Graphing

We will use the same problem solving strategy we used in Math Models to set up and solve applications of systems of linear equations. We’ll modify the strategy slightly here to make it appropriate for systems of equations.

USE A PROBLEM SOLVING STRATEGY FOR SYSTEMS OF LINEAR EQUATIONS.

• Read the problem. Make sure all the words and ideas are understood.
• Identify what we are looking for.
• Name what we are looking for. Choose variables to represent those quantities.
• Translate into a system of equations.
• Solve the system of equations using good algebra techniques.
• Check the answer in the problem and make sure it makes sense.
• Answer the question with a complete sentence.

Step 5 is where we will use the method introduced in this section. We will graph the equations and find the solution.

Example \(\PageIndex{34}\)

Sondra is making 10 quarts of punch from fruit juice and club soda. The number of quarts of fruit juice is 4 times the number of quarts of club soda. How many quarts of fruit juice and how many quarts of club soda does Sondra need?

Step 1. Read the problem.

Step 2. Identify what we are looking for.

We are looking for the number of quarts of fruit juice and the number of quarts of club soda that Sondra will need.

Step 3. Name what we are looking for. Choose variables to represent those quantities.

  Let f= number of quarts of fruit juice.     c= number of quarts of club soda

Step 4. Translate into a system of equations.

We now have the system. \(\begin{cases}{ f+c=10} \\ {f=4c}\end{cases}\)

Step 5. Solve the system of equations using good algebra techniques.

The point of intersection (2, 8) is the solution. This means Sondra needs 2 quarts of club soda and 8 quarts of fruit juice.

Step 6. Check the answer in the problem and make sure it makes sense.

Does this make sense in the problem?

Yes, the number of quarts of fruit juice, 8 is 4 times the number of quarts of club soda, 2.

Yes, 10 quarts of punch is 8 quarts of fruit juice plus 2 quarts of club soda.

Step 7. Answer the question with a complete sentence.

Sondra needs 8 quarts of fruit juice and 2 quarts of soda.

Try It \(\PageIndex{35}\)

Manny is making 12 quarts of orange juice from concentrate and water. The number of quarts of water is 3 times the number of quarts of concentrate. How many quarts of concentrate and how many quarts of water does Manny need?

Manny needs 3 quarts juice concentrate and 9 quarts water.

Try It \(\PageIndex{36}\)

Alisha is making an 18 ounce coffee beverage that is made from brewed coffee and milk. The number of ounces of brewed coffee is 5 times greater than the number of ounces of milk. How many ounces of coffee and how many ounces of milk does Alisha need?

Alisha needs 15 ounces of coffee and 3 ounces of milk.

Access these online resources for additional instruction and practice with solving systems of equations by graphing.

• Instructional Video Solving Linear Systems by Graphing
• Instructional Video Solve by Graphing

Key Concepts

• Identify the solution to the system. If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions.
• Check the solution in both equations.

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Chapter 3: Graphing

3.4 Graphing Linear Equations

There are two common procedures that are used to draw the line represented by a linear equation. The first one is called the slope-intercept method and involves using the slope and intercept given in the equation.

If the equation is given in the form [latex]y = mx + b[/latex], then [latex]m[/latex] gives the rise over run value and the value [latex]b[/latex] gives the point where the line crosses the [latex]y[/latex]-axis, also known as the [latex]y[/latex]-intercept.

Example 3.4.1

Given the following equations, identify the slope and the [latex]y[/latex]-intercept.

• [latex]\begin{array}{lll} y = 2x - 3\hspace{0.14in} & \text{Slope }(m)=2\hspace{0.1in}&y\text{-intercept } (b)=-3 \end{array}[/latex]
• [latex]\begin{array}{lll} y = \dfrac{1}{2}x - 1\hspace{0.08in} & \text{Slope }(m)=\dfrac{1}{2}\hspace{0.1in}&y\text{-intercept } (b)=-1 \end{array}[/latex]
• [latex]\begin{array}{lll} y = -3x + 4 & \text{Slope }(m)=-3 &y\text{-intercept } (b)=4 \end{array}[/latex]
• [latex]\begin{array}{lll} y = \dfrac{2}{3}x\hspace{0.34in} & \text{Slope }(m)=\dfrac{2}{3}\hspace{0.1in} &y\text{-intercept } (b)=0 \end{array}[/latex]

When graphing a linear equation using the slope-intercept method, start by using the value given for the [latex]y[/latex]-intercept. After this point is marked, then identify other points using the slope.

This is shown in the following example.

Example 3.4.2

Graph the equation [latex]y = 2x - 3[/latex].

First, place a dot on the [latex]y[/latex]-intercept, [latex]y = -3[/latex], which is placed on the coordinate [latex](0, -3).[/latex]

Now, place the next dot using the slope of 2.

A slope of 2 means that the line rises 2 for every 1 across.

Simply, [latex]m = 2[/latex] is the same as [latex]m = \dfrac{2}{1}[/latex], where [latex]\Delta y = 2[/latex] and [latex]\Delta x = 1[/latex].

Placing these points on the graph becomes a simple counting exercise, which is done as follows:

Once several dots have been drawn, draw a line through them, like so:

Note that dots can also be drawn in the reverse of what has been drawn here.

Slope is 2 when rise over run is [latex]\dfrac{2}{1}[/latex] or [latex]\dfrac{-2}{-1}[/latex], which would be drawn as follows:

Example 3.4.3

Graph the equation [latex]y = \dfrac{2}{3}x[/latex].

First, place a dot on the [latex]y[/latex]-intercept, [latex](0, 0)[/latex].

Now, place the dots according to the slope, [latex]\dfrac{2}{3}[/latex].

This will generate the following set of dots on the graph. All that remains is to draw a line through the dots.

The second method of drawing lines represented by linear equations and functions is to identify the two intercepts of the linear equation. Specifically, find [latex]x[/latex] when [latex]y = 0[/latex] and find [latex]y[/latex] when [latex]x = 0[/latex].

Example 3.4.4

Graph the equation [latex]2x + y = 6[/latex].

To find the first coordinate, choose [latex]x = 0[/latex].

This yields:

[latex]\begin{array}{lllll} 2(0)&+&y&=&6 \\ &&y&=&6 \end{array}[/latex]

Coordinate is [latex](0, 6)[/latex].

Now choose [latex]y = 0[/latex].

[latex]\begin{array}{llrll} 2x&+&0&=&6 \\ &&2x&=&6 \\ &&x&=&\frac{6}{2} \text{ or } 3 \end{array}[/latex]

Coordinate is [latex](3, 0)[/latex].

Draw these coordinates on the graph and draw a line through them.

Example 3.4.5

Graph the equation [latex]x + 2y = 4[/latex].

[latex]\begin{array}{llrll} (0)&+&2y&=&4 \\ &&y&=&\frac{4}{2} \text{ or } 2 \end{array}[/latex]

Coordinate is [latex](0, 2)[/latex].

[latex]\begin{array}{llrll} x&+&2(0)&=&4 \\ &&x&=&4 \end{array}[/latex]

Coordinate is [latex](4, 0)[/latex].

Example 3.4.6

Graph the equation [latex]2x + y = 0[/latex].

[latex]\begin{array}{llrll} 2(0)&+&y&=&0 \\ &&y&=&0 \end{array}[/latex]

Coordinate is [latex](0, 0)[/latex].

Since the intercept is [latex](0, 0)[/latex], finding the other intercept yields the same coordinate. In this case, choose any value of convenience.

Choose [latex]x = 2[/latex].

[latex]\begin{array}{rlrlr} 2(2)&+&y&=&0 \\ 4&+&y&=&0 \\ -4&&&&-4 \\ \hline &&y&=&-4 \end{array}[/latex]

Coordinate is [latex](2, -4)[/latex].

For questions 1 to 10, sketch each linear equation using the slope-intercept method.

• [latex]y = -\dfrac{1}{4}x - 3[/latex]
• [latex]y = \dfrac{3}{2}x - 1[/latex]
• [latex]y = -\dfrac{5}{4}x - 4[/latex]
• [latex]y = -\dfrac{3}{5}x + 1[/latex]
• [latex]y = -\dfrac{4}{3}x + 2[/latex]
• [latex]y = \dfrac{5}{3}x + 4[/latex]
• [latex]y = \dfrac{3}{2}x - 5[/latex]
• [latex]y = -\dfrac{2}{3}x - 2[/latex]
• [latex]y = -\dfrac{4}{5}x - 3[/latex]
• [latex]y = \dfrac{1}{2}x[/latex]

For questions 11 to 20, sketch each linear equation using the [latex]x\text{-}[/latex] and [latex]y[/latex]-intercepts.

• [latex]x + 4y = -4[/latex]
• [latex]2x - y = 2[/latex]
• [latex]2x + y = 4[/latex]
• [latex]3x + 4y = 12[/latex]
• [latex]4x + 3y = -12[/latex]
• [latex]x + y = -5[/latex]
• [latex]3x + 2y = 6[/latex]
• [latex]x - y = -2[/latex]
• [latex]4x - y = -4[/latex]

For questions 21 to 28, sketch each linear equation using any method.

• [latex]y = -\dfrac{1}{2}x + 3[/latex]
• [latex]y = 2x - 1[/latex]
• [latex]y = -\dfrac{5}{4}x[/latex]
• [latex]y = -3x + 2[/latex]
• [latex]y = -\dfrac{3}{2}x + 1[/latex]
• [latex]y = \dfrac{1}{3}x - 3[/latex]
• [latex]y = \dfrac{3}{2}x + 2[/latex]
• [latex]y = 2x - 2[/latex]

For questions 29 to 40, reduce and sketch each linear equation using any method.

• [latex]y + 3 = -\dfrac{4}{5}x + 3[/latex]
• [latex]y - 4 = \dfrac{1}{2}x[/latex]
• [latex]x + 5y = -3 + 2y[/latex]
• [latex]3x - y = 4 + x - 2y[/latex]
• [latex]4x + 3y = 5 (x + y)[/latex]
• [latex]3x + 4y = 12 - 2y[/latex]
• [latex]2x - y = 2 - y \text{ (tricky)}[/latex]
• [latex]7x + 3y = 2(2x + 2y) + 6[/latex]
• [latex]x + y = -2x + 3[/latex]
• [latex]3x + 4y = 3y + 6[/latex]
• [latex]2(x + y) = -3(x + y) + 5[/latex]
• [latex]9x - y = 4x + 5[/latex]

Answer Key 3.4

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

Share This Book

Free Printable Math Worksheets for Algebra 2

Created with infinite algebra 2, stop searching. create the worksheets you need with infinite algebra 2..

• Fast and easy to use
• Multiple-choice & free-response
• Never runs out of questions
• Multiple-version printing

Free 14-Day Trial

• Order of operations
• Evaluating expressions
• Simplifying algebraic expressions
• Multi-step equations
• Work word problems
• Distance-rate-time word problems
• Mixture word problems
• Absolute value equations
• Multi-step inequalities
• Compound inequalities
• Absolute value inequalities
• Discrete relations
• Continuous relations
• Evaluating and graphing functions
• Review of linear equations
• Graphing absolute value functions
• Graphing linear inequalities
• Direct and inverse variation
• Systems of two linear inequalities
• Systems of two equations
• Systems of two equations, word problems
• Points in three dimensions
• Systems of three equations, elimination
• Systems of three equations, substitution
• Basic matrix operations
• Matrix multiplication
• All matrix operations combined
• Matrix inverses
• Geometric transformations with matrices
• Operations with complex numbers
• Properties of complex numbers
• Rationalizing imaginary denominators
• Properties of parabolas
• Vertex form
• Graphing quadratic inequalities
• Factoring quadratic expressions
• Solving quadratic equations w/ square roots
• Solving quadratic equations by factoring
• Completing the square
• Solving equations by completing the square
• Solving equations with the quadratic formula
• The discriminant
• Naming and simple operations
• Factoring a sum/difference of cubes
• Factoring by grouping
• Factoring quadratic form
• Factoring using all techniques
• Factors and Zeros
• The Remainder Theorem
• Irrational and Imaginary Root Theorems
• Descartes' Rule of Signs
• More on factors, zeros, and dividing
• The Rational Root Theorem
• Polynomial equations
• Basic shape of graphs of polynomials
• Graphing polynomial functions
• The Binomial Theorem
• Evaluating functions
• Function operations
• Inverse functions
• Simplifying radicals
• Operations with radical expressions
• Dividing radical expressions
• Radicals and rational exponents
• Simplifying rational exponents
• Square root equations
• Rational exponent equations
• Graphing radicals
• Graphing & properties of parabolas
• Equations of parabolas
• Graphing & properties of circles
• Equations of circles
• Graphing & properties of ellipses
• Equations of ellipses
• Graphing & properties of hyperbolas
• Equations of hyperbolas
• Classifying conic sections
• Eccentricity
• Systems of quadratic equations
• Graphing simple rational functions
• Graphing general rational functions
• Simplifying rational expressions
• Multiplying / dividing rational expressions
• Adding / subtracting rational expressions
• Complex fractions
• Solving rational equations
• The meaning of logarithms
• Properties of logarithms
• The change of base formula
• Writing logs in terms of others
• Logarithmic equations
• Inverse functions and logarithms
• Exponential equations not requiring logarithms
• Exponential equations requiring logarithms
• Graphing logarithms
• Graphing exponential functions
• Discrete exponential growth and decay word problems
• Continuous exponential growth and decay word problems
• General sequences
• Arithmetic sequences
• Geometric sequences
• Comparing Arithmetic/Geometric Sequences
• General series
• Arithmetic series
• Arithmetic/Geometric Means w/ Sequences
• Finite geometric series
• Infinite geometric series
• Right triangle trig: Evaluating ratios
• Right triangle trig: Missing sides/angles
• Angles and angle measure
• Co-terminal angles and reference angles
• Arc length and sector area
• Trig ratios of general angles
• Exact trig ratios of important angles
• The Law of Sines
• The Law of Cosines
• Graphing trig functions
• Translating trig functions
• Angle Sum/Difference Identities
• Double-/Half-Angle Identities
• Sample spaces and The Counting Principle
• Independent and dependent events
• Mutualy exclusive events
• Permutations
• Combinations
• Permutations vs combinations
• Probability using permutations and combinations

Graphing Parabolas Worksheets

How to Graph a Parabola - A parabola is a mirror-symmetrical and a two-dimensional curve that resembles the shape of an arch. Any point on the parabola is present equidistant from the focus and fixed straight line known as the directrix. For graphing parabola, we need to figure out the vertex and other points on sides of vertex to find out the path a point travel. Here we have discussed the steps required for graphing a parabola. First, identify the relevant parts of the parabola. Then figure out the equation of the parabola. If the coefficient in the equation is positive, the parabola opens upward, and if the coefficient is negative, the parabola opens downward. If the equation of a parabola has squared y term instead of squared x term, the parabola will open sideways and oriented horizontally. Figure out the axis of symmetry passing through the vertex. In vertical parabola, the axis of symmetry is the same as the x-axis. That means, the value of x where a parabola crosses the axis of symmetry. Once you have found out the axis of symmetry, put the values in to get the values of x and y coordinates. Create a table by using the chosen x values and write them down in the first column. The tables will give you the coordinates required to graph the equation. Plug in the values of x in the equation to get the values of the corresponding y values. Next, insert the values of y in the table. Graph the x and y values on the graph. Connect the points that you plotted to form the graph of the parabola.

Basic Lesson

Demonstrates how to determine the pitch of a parabola prior to graphing it. Practice problems are provided. Parabolas are of quadratic form y= ax 2 + bx +c If a is positive, the parabola will open upward and has minimum point. If a is negative, the parabola will open downward and has maximum point.

Intermediate Lesson

Explores how to graph parabolas within a given interval. Practice problems are provided. Graph the parabola y = x 2 - 4x on interval (-1, 5). Place the value of x {-1, 0, 1, 2, 3, 4, 5} in the equation and calculate y.

Independent Practice 1

Find the roots of following graphs. The answers can be found below.

Independent Practice 2

Examples: What is the equation of the axis of symmetry for parabola with a turning point of (1, 6)? Does the quadratic equation (parabola) y= -x 2 -6x + 2 have axis of symmetry where x = 3?

Homework Worksheet

12 Graphing Parabolas problems for students to work on at home. Example problems are provided and explained.

10 Graphing Parabolas problems. A math scoring matrix is included.

Homework and Quiz Answer Key

Answers for the homework and quiz.

Lesson and Practice Answer Key

Answers for both lessons and both practice sheets.

Demonstrates how to determine the pitch and axis of symmetry for a parabola. Will the graph of the parabola y = 5x 2 + 3x - 8 open upward or downward? Consider the equation of a parabola in the form of y = ax 2 + bx + c If 'a' value is positive, it opens upward and vice versa.

Explores how to determine the coefficient 'a' and roots of a parabola. The coefficient'a' of the parabola is positive or negative? As the graph opens upward, coefficient 'a' value is positive.

What is the turning point of the parabola y = -x 2 + 8x – 3? The answers can be found below.

Features another 20 Graphs of Parabolas problems.

Graphs of Parabolas problems for students to work on at home. Example problems are provided and explained.

10 Graphs of Parabolas problems. A math scoring matrix is included.

How to Predict Movement of a Graph of a Parabola

To determine the shift or movement in the graph of a parabola, we need to find out its vertex and replot the point. And most importantly, we will have to learn to read the equation and understand how the equation shifts vertically and horizontally. Let's understand this with the simplest parabola equation y= x 2 . The equation has the vertex at (0,0) and opens upward. Assuming we consider the points (-1, 1), (1,1), (-2,4) and (2,4,). We can understand the shifting based on this equation. If we want to shift parabola upward, we will consider the equation y= x 2 + 1. This way, we can shift the parabola upward by 1 unit. The vertex will now shift to (0,1) from (0,0). It will have the same shape as the original parabola, but the y-coordinates will be moved upward to 1 unit. If we want to shift the parabola downward, we will take the equation y= x 2 - 1. This way, we shift the original parabola downward by 1 unit. The vertex now moves to (0,-1) from (0,0). The parabola will have the same shape as the original parabola, but y-coordinates will be shifted downward by 1 unit.

The Designs

A mathematician is asked to design a table. He first designs a table with no legs. Then he designs a table with infinitely many legs. He spend the rest of his life generalizing the results for the table with N legs (where N is not necessarily a natural number).

A Little About Archimedes

Archimedes (287-212 BC) was a Greek mathematician who discovered the formulae for finding the areas and volume of spheres, cylinders, parabolas and other plane and solid figures. Known for crying "eureka" when he discovered the buoyancy principle.

Addition (Basic)

Addition (Multi-Digit)

Algebra & Pre-Algebra

Comparing Numbers

Daily Math Review

Division (Basic)

Division (Long Division)

Hundreds Charts

Measurement

Multiplication (Basic)

Multiplication (Multi-Digit)

Order of Operations

Place Value

Probability

Skip Counting

Subtraction

Telling Time

Word Problems (Daily)

More Math Worksheets

Reading Comprehension

Reading Comprehension Gr. 1

Reading Comprehension Gr. 2

Reading Comprehension Gr. 3

Reading Comprehension Gr. 4

Reading Comprehension Gr. 5

Reading Comprehension Gr. 6

Reading & Writing

Reading Worksheets

Cause & Effect

Fact & Opinion

Fix the Sentences

Graphic Organizers

Synonyms & Antonyms

Writing Prompts

Writing Story Pictures

Writing Worksheets

More ELA Worksheets

Consonant Sounds

Vowel Sounds

Consonant Blends

Consonant Digraphs

Word Families

More Phonics Worksheets

Early Literacy

Build Sentences

Sight Word Units

Sight Words (Individual)

More Early Literacy

Punctuation

Subjects and Predicates

More Grammar Worksheets

Spelling Lists

Spelling Grade 1

Spelling Grade 2

Spelling Grade 3

Spelling Grade 4

Spelling Grade 5

Spelling Grade 6

More Spelling Worksheets

Chapter Books

Charlotte's Web

Magic Tree House #1

Boxcar Children

More Literacy Units

Animal (Vertebrate) Groups

Butterfly Life Cycle

Electricity

Matter (Solid, Liquid, Gas)

Simple Machines

Space - Solar System

More Science Worksheets

Social Studies

Maps (Geography)

Maps (Map Skills)

More Social Studies

Mother's Day

Father's Day

More Holiday Worksheets

Puzzles & Brain Teasers

Brain Teasers

Logic:  Addition Squares

Mystery Graph Pictures

Number Detective

Lost in the USA

More Thinking Puzzles

Teacher Helpers

Teaching Tools

Award Certificates

More Teacher Helpers

Pre-K and Kindergarten

Alphabet (ABCs)

Numbers and Counting

Shapes (Basic)

More Kindergarten

Worksheet Generator

Word Search Generator

Multiple Choice Generator

Fill-in-the-Blanks Generator

More Generator Tools

Full Website Index

Our collection of histogram worksheets helps students learn how to read and create this type of graph. Using given data, students can fill in histograms on their own and answer questions interpreting them.

Logged in members can use the Super Teacher Worksheets filing cabinet to save their favorite worksheets.

Quickly access your most used files AND your custom generated worksheets!

Please login to your account or become a member and join our community today to utilize this helpful feature.

Learn to read and interpret data on bar graphs with these printable PDFs.

Practice reading and creating bar graphs, line graphs, line plots, pictographs, and pie graphs.

Sample Worksheet Images

PDF with answer key:

PDF no answer key:

Check Out the New Website Shop!

Novels & Picture Books

Anchor Charts

• Geometry & Measurement
• Graphs and Data
• Math Anchor Charts

Patterns & Coordinate Graphing

By Mary Montero

Share This Post:

• Facebook Share
• Twitter Share
• Pinterest Share
• Email Share

We are finishing up our patterns and coordinate graphing unit!  My kids had so much fun with this one, especially coming up with their own patterns.  They love to really push the limits and think outside the box when creating their patterns.

We began the unit with patterns and function tables.  We did both shape and number patterns–all kinds of patterns.  On the more complicated number patterns that have alternating rules or less than obvious patterns, I teach kids to use a bridge underneath each number to recognize what is happening.  They look at it in all directions– are they adding, subtracting, multiplying, or dividing each number?  Then they learn to look at the pattern within the bridges.  I have seen year after year that kids use the bridges on their work, especially on state testing.

We used a differentiated packet that I created to practice patterning.  We did all kinds of practice, and the kids actually wanted MORE.

After we had done plenty of practice with the skill, I had my students work through several Patterns and Functions Task Cards .  I created this set to have a wide variety of different skills practiced throughout.  This time around, we did a simple task card scavenger hunt, where I had the cards hung up all around the hallway and in the classroom, and they completed them at each location.  It’s super simple prep, but the kids love getting up and moving around!

After I felt like they had mastered this concept, I moved on to coordinate graphing, which was a breeze for my students.  They copied the anchor chart in their math journal and then did a proof.  We practiced with a bunch of function tables.  We created coordinate graph pictures of our own (which they LOVED creating) and worked through a few story problems.  This only took a day or two and they were solid!

This week, they will be taking their quiz over this unit, which I offer for FREE at my TpT store.  Click on the picture or here to download it for free.  (An answer key is included, too!)

Mary Montero

I’m so glad you are here. I’m a current gifted and talented teacher in a small town in Colorado, and I’ve been in education since 2009. My passion (other than my family and cookies) is for making teachers’ lives easier and classrooms more engaging.

You might also like…

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Great post! Love the patterns poster at the top!!

This comment has been removed by a blog administrator.

©2023 Teaching With a Mountain View . All Rights Reserved | Designed by Ashley Hughes

Username or Email Address

Remember Me

Lost your password?

Review Cart

No products in the cart.

Functions and Graphing (Math 7 Curriculum - Unit 5) | All Things Algebra®

• Google Apps™

What educators are saying

Also included in.

Description

This Functions and Graphing Unit Bundle includes guided notes, homework assignments, two quizzes, a study guide, and a unit test that cover the following topics:

• The Coordinate Plane

• Graphing Relations

• Graphing Functions using a Function Table

• Multiple Representations of Functions: Tables, Graphs, Equations, Verbal Descriptions

• Slope from a Graph (positive and negative slope only)

• Slope-Intercept Form

• Graphing a Linear Function in Slope-Intercept Form

• Slope-Intercept Form Applications

• Proportional Relationships (recognizing a proportional relationship given a table or graph, identify the constant of proportionality given a table or graph, write an equation to represent a proportional relationship, graph a proportional relationship)

ADDITIONAL COMPONENTS INCLUDED:

(1) Links to Instructional Videos: Links to videos of each lesson in the unit are included. Videos were created by fellow teachers for their students using the guided notes and shared in March 2020 when schools closed with no notice.  Please watch through first before sharing with your students. Many teachers still use these in emergency substitute situations. (2) Editable Assessments: Editable versions of each quiz and the unit test are included. PowerPoint is required to edit these files. Individual problems can be changed to create multiple versions of the assessment. The layout of the assessment itself is not editable. If your Equation Editor is incompatible with mine (I use MathType), simply delete my equation and insert your own.

(3) Google Slides Version of the PDF: The second page of the Video links document contains a link to a Google Slides version of the PDF. Each page is set to the background in Google Slides. There are no text boxes;  this is the PDF in Google Slides.  I am unable to do text boxes at this time but hope this saves you a step if you wish to use it in Slides instead! 

This resource is included in the following bundle(s):

Math 7 Curriculum

More Math 7 Units:

Unit 1 – Number Sense

Unit 2 – Expressions

Unit 3 – Equations and Inequalities

Unit 4 – Ratios, Proportions, and Percents

Unit 6 – Geometry

Unit 7 – Measurement: Area and Volume

Unit 8 – Probability and Statistics

LICENSING TERMS: This purchase includes a license for one teacher only for personal use in their classroom. Licenses are non-transferable , meaning they can not be passed from one teacher to another. No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses. If you are a coach, principal, or district interested in transferable licenses to accommodate yearly staff changes, please contact me for a quote at [email protected].

COPYRIGHT TERMS: This resource may not be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.

© All Things Algebra (Gina Wilson), 2012-present

Questions & Answers

All things algebra.

• We're hiring
• Help & FAQ
• Privacy policy
• Student privacy
• Terms of service
• Tell us what you think

• English Language Arts
• Graphic Organizers
• Social Studies
• Teacher Printables
• Foreign Language

Home > Math Worksheets > Algebra Worksheets > Direct Variations

Whenever you have two variables that share in a simple relationship, they can take the form of a direct variation. We can say that one variable varies directly with another variable. This usually happens when one variable increases or decreases and then the related variable that we defined does as well. These lessons and worksheets help your students to understand the concept of direct variation with regard to equations with variables. These are a set of unique worksheets that students can add to their collection of activity sheets. They will help them learn how to solve for variables in different algebraic expressions by using direct variation. Students will also use grids in order to determine the relationships of variables.

Get Free Worksheets In Your Inbox!

Print direct variation worksheets, click the buttons to print each worksheet and associated answer key., direct variation lesson.

Learn how to solve problems like the following: x varies directly with y. If x = 4 when y = 14, find y when x = 2

Direct Variation Worksheet 1

You will solve these word problems dealing with direct variation between variables. Example: The distance of a train from a station, varies directly with the time, t. If d = 100 miles when t = 2 hours, find d when t = 3.

Worksheet 2

The problems on this worksheet are much different than worksheet 1. You will find a mix of word and number based problems. Example: If x varies directly with y and x is 21 when y is 10, find the constant of variation.

Review Sheet

Review the steps to solving equations dealing with direct variation between variables. Example: In the following chart, does one variable vary directly with the other.

Practice Worksheet

Practice solving these direct variation problems that are all number based. Example: p varies directly with q". If p = 3 when q = 21, find p when q = 3.

Solve these 10 problems and then score how many answers you got correct. Example: If x varies directly with y and x is 24 when y is 10, find the constant of variation. This is a great way to see how much work you might or might not need to put in.

Introduction / Review Sheet

Complete these 3 problems, then put your answer in the "My Answer" box for each. This can help you review or introduce this skill to students.

What are Direct Variations in Math?

Direct variations are an integral part of mathematics you regularly encounter when studying. Many people find direct variations to be confusing. If you are one of them, read here to learn what direct variations are and how to solve them.

Direct variations are a category of proportionality. They show how one variable relates varies with another. Direct variation indicates a linear relationship between two variables. Direct variation is also called direct proportionality.

Two variables that increase and decrease by the same amount are directly proportional. If one quantity increases or decreases, the other follows suit.

For example, the height of the wall is directly proportional to the number of bricks. If the number of bricks in the wall increase, so does the height.

Direct variation is formulated using the symbol ‘∝,’ which shows that two values follow direct variation. For example, if two values, x, and y, are directly proportional, they are expressed as y ∝ x.

The two values, y, and x, increase and decrease by the same factor. This factor is constant. This means that it will not change even if the values themselves do. We can denote this factor as the constant 'k.' so the formula y ∝ x becomes y =kx or x = y/k (where the constant becomes 1/k).

Solving Direct Variation Problems

Solving direct variation problems is easy once you understand the formula. Since the constant does not change, you need to figure out the constant, and the values of y or x can be found, given that one of the values is given. To better understand the concept, let’s look at some examples:

Example 1: The number of cookies made varies directly with the flour used. If 12 cookies require 2 cups of flour, how many cookies can you make with 6 cups of flour?

Let y = number of cookies made and x = flour used. Since the two are directly proportional, we can write them as

to find k, we use y = 12 and x= 2

12 = k (2) k = 12/6 k = 2.

Now we used this value of k and the x = 6 to find y,

y = kx y = 2(6) y = 12

Twelve cookies can be made using six cups of flour.

Example 2: The number of cookies made varies directly with the flour used. If making 36 cookies requires 3 cups of flour, how much flour would you need to make 72 cookies?

Let y = number of cookies and x = flour used. Since the two are directly proportional, we can write them as,

we use y = 36 and x = 3 to find the value of k,

36 = k (3) k = 12 now, we use the value of k and y = 72 to find x, y = kx x = y/k x = 72/12 x = 6

You would need six cups of flour to make seventy-two cookies.

Direct variations are easy to do once you get the concept. But understanding the concept only gets you halfway to mastering direct variation questions. The other half you will cover after practicing these questions. Practice makes perfect!