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solving systems of equations by elimination word problems

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Unit 6: Lesson 6

Systems of equations with elimination: TV & DVD

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Video transcript

ELIMINATION METHOD WORD PROBLEMS WORKSHEET

Problem 1 :

A park charges $10 for adults and $5 for kids. How many many adults tickets and kids tickets were sold, if a total of 548 tickets were sold for a total of $3750 ? 

Problem 2 :

Sum of the cost price of two products is $50. Sum of the selling price of the same two products is $52. If one is sold at 20% profit and other one is sold at 20% loss, find the cost price of each product.  

solving systems of equations by elimination word problems

Detailed Answer Key

Let x be the no. of adult tickets and y be the no. of kids tickets.

According to the question, we have

x + y  =  548 ---------(1)

10x + 5y  =  3750

Divide both sides by 5.

2x + y  =  750 --------(2)

Step 2 : 

Eliminate one of the variables to get the value of the other variable.

In (1) and (2), variable y is having the same coefficient. But, the variable y is having the same sign in both the equations. 

To change the sign of y in (1), multiply both sides of (1) by negative sign.

- (x + y)  =  - 548

- x - y  =  - 548  --------(3)

Step 3 : 

Now, eliminate the variable y in (2) and (3) as given below and find the value of x. 

solving systems of equations by elimination word problems

Step 4 : 

Substitute 202 for x in (1) to get the value of y. 

               (2) -------->  202 + y  =  548 

Subtract 202 from both sides. 

y  =  346

So, the number of adults tickets sold is 202 and the number of kids tickets sold is 346.

Let x and "y" be the cost prices of two products. 

Then,  x + y  =  50  --------(1)

Let us assume that x is sold at 20% profit

Then, the selling price of x is 120% of x.

Selling price of x  =  1.2x

Let us assume that y is sold at 20% loss

Then, the selling price of y is 80% of y.

Selling price of x  =  0.8y

Given : Selling price of x + Selling price of y  =  52

1.2x + 0.8y  =  52

To avoid decimal, multiply both sides by 10

12x + 8y  =  520

Divide both sides by 4.

3x + 2y  =  130 --------(2)

In (1) and (2), both the variables x and y are not having the same coefficient.

One of the variables must have the same coefficient. 

So multiply both sides of (1) by 2 to make the coefficients of y same in both the equations.  

(1)  ⋅  2 -------->  2x + 2y  =  100 ----------(3) 

Variable y is having the same sign in both (2) and (3). 

To change the sign of y in (3), multiply both sides of (3) by negative sign.

- (2x + 2y)  =  - 100

- 2x - 2y  =  - 100 --------(4)

Now, eliminate the variable y in (2) and (4) as given below and find the value of x. 

solving systems of equations by elimination word problems

Step 5 : 

Substitute 30 for x  in (1) to get the value of y. 

                    (2) --------> 30  + y  =  50

Subtract 30 from both sides.

y  =  20

So,  the cost prices of two products are $30 and $20.

solving systems of equations by elimination word problems

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Solving Systems of Equations Real World Problems

Wow! You have learned many different strategies for solving systems of equations! First we started with Graphing Systems of Equations . Then we moved onto solving systems using the Substitution Method . In our last lesson we used the Linear Combinations or Addition Method to solve systems of equations.

Now we are ready to apply these strategies to solve real world problems! Are you ready? First let's look at some guidelines for solving real world problems and then we'll look at a few examples.

Steps For Solving Real World Problems

Ok... let's look at a few examples. Follow along with me. (Having a calculator will make it easier for you to follow along.)

Example 1: Systems Word Problems

You are running a concession stand at a basketball game. You are selling hot dogs and sodas. Each hot dog costs $1.50 and each soda costs $0.50. At the end of the night you made a total of $78.50. You sold a total of 87 hot dogs and sodas combined. You must report the number of hot dogs sold and the number of sodas sold. How many hot dogs were sold and how many sodas were sold?

1.  Let's start by identifying the important information:

2.  Define your variables.

In this problem, I don't know how many hot dogs or sodas were sold. So this is what each variable will stand for. (Usually the question at the end will give you this information).

Let x = the number of hot dogs sold

Let y = the number of sodas sold

3. Write two equations.

One equation will be related to the price and one equation will be related to the quantity (or number) of hot dogs and sodas sold.

1.50x + 0.50y = 78.50    (Equation related to cost)

 x + y = 87   (Equation related to the number sold)

4.  Solve! 

We can choose any method that we like to solve the system of equations. I am going to choose the substitution method since I can easily solve the 2nd equation for y.

Solving a systems using substitution

5. Think about what this solution means.

x is the number of hot dogs and x = 35. That means that 35 hot dogs were sold.

y is the number of sodas and y = 52. That means that 52 sodas were sold.

6.  Write your answer in a complete sentence.

35 hot dogs were sold and 52 sodas were sold.

7.  Check your work by substituting.

1.50x + 0.50y = 78.50

1.50(35) + 0.50(52) = 78.50

52.50 + 26 = 78.50

35 + 52 = 87

Since both equations check properly, we know that our answers are correct!

That wasn't too bad, was it? The hardest part is writing the equations. From there you already know the strategies for solving. Think carefully about what's happening in the problem when trying to write the two equations.

Example 2: Another Word Problem

You and a friend go to Tacos Galore for lunch. You order three soft tacos and three burritos and your total bill is $11.25. Your friend's bill is $10.00 for four soft tacos and two burritos. How much do soft tacos cost? How much do burritos cost?

In this problem, I don't know the price of the soft tacos or the price of the burritos.

Let x = the price of 1 soft taco

Let y = the price of 1 burrito

One equation will be related your lunch and one equation will be related to your friend's lunch.

3x + 3y = 11.25  (Equation representing your lunch)

4x + 2y = 10   (Equation representing your friend's lunch)

We can choose any method that we like to solve the system of equations. I am going to choose the combinations method.

Solving Systems Using Combinations

5. Think about what the solution means in context of the problem.

x = the price of 1 soft taco and x = 1.25.

That means that 1 soft tacos costs $1.25.

y = the price of 1 burrito and y = 2.5.

That means that 1 burrito costs $2.50.

Yes, I know that word problems can be intimidating, but this is the whole reason why we are learning these skills. You must be able to apply your knowledge!

If you have difficulty with real world problems, you can find more examples and practice problems in the Algebra Class E-course.

Take a look at the questions that other students have submitted:

solving systems of equations by elimination word problems

Problem about the WNBA

Systems problem about ages

Problem about milk consumption in the U.S.

Vans and Buses? How many rode in each?

Telephone Plans problem

Systems problem about hats and scarves

Apples and guavas please!

How much did Alice spend on shoes?

All about stamps

Going to the movies

Small pitchers and large pitchers - how much will they hold?

Chickens and dogs in the farm yard

solving systems of equations by elimination word problems

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Systems of Linear Equations Word Problems: Elimination

Students practice solving word problems by writing and solving systems of equations in this eighth-grade algebra worksheet! For each problem in this two-page worksheet, students are asked to write and solve a system of equations using the elimination method. Systems of Linear Equations Word Problems: Elimination will give students practice writing equations to model real-world problems and solving systems of equations using the elimination method. For more practice, have learners complete the Systems of Linear Equations Word Problems: Graphing and Systems of Linear Equations Word Problems: Substitution worksheets.

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