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Eureka Math Grade 8 Module 4 Lesson 1 Answer Key

Engage ny eureka math 8th grade module 4 lesson 1 answer key, eureka math grade 8 module 4 lesson 1 exercise answer key.

Write each of the following statements using symbolic language.

Exercise 1. The sum of four consecutive even integers is -28. Answer: Let x be the first even integer. Then, x+x+2+x+4+x+6=-28.

Exercise 2. A number is four times larger than the square of half the number. Answer: Let x be the number. Then, x=4(\(\frac{x}{2}\)) 2 .

Exercise 3. Steven has some money. If he spends $9.00, then he will have \(\frac{3}{5}\) of the amount he started with. Answer: Let x be the amount of money (in dollars) Steven started with. Then, x-9=\(\frac{3}{5}\) x.

Exercise 4. The sum of a number squared and three less than twice the number is 129. Answer: Let x be the number. Then, x 2 +2x-3=129.

Exercise 5. Miriam read a book with an unknown number of pages. The first week, she read five less than \(\frac{1}{3}\) of the pages. The second week, she read 171 more pages and finished the book. Write an equation that represents the total number of pages in the book. Answer: Let x be the total number of pages in the book. Then, \(\frac{1}{3}\) x-5+171=x.

Eureka Math Grade 8 Module 4 Lesson 1 Problem Set Answer Key

Students practice transcribing written statements into symbolic language.

Question 1. Bruce bought two books. One book costs $4.00 more than three times the other. Together, the two books cost him $72. Answer: Let x be the cost of the less expensive book. Then, x+4+3x=72.

Question 2. Janet is three years older than her sister Julie. Janet’s brother is eight years younger than their sister Julie. The sum of all of their ages is 55 years. Answer: Let x be Julie’s age. Then, (x+3)+(x-8)+x=55.

Question 3. The sum of three consecutive integers is 1,623. Answer: Let x be the first integer. Then, x+(x+1)+(x+2)=1623.

Question 4. One number is six more than another number. The sum of their squares is 90. Answer: Let x be the smaller number. Then, x 2 +(x+6) 2 =90.

Question 5. When you add 18 to \(\frac{1}{4}\) of a number, you get the number itself. Answer: Let x be the number. Then, \(\frac{1}{4}\) x+18=x.

Question 6. When a fraction of 17 is taken away from 17, what remains exceeds one-third of seventeen by six. Answer: Let x be the fraction of 17. Then, 17-x=\(\frac{1}{3}\)∙17+6.

Question 7. The sum of two consecutive even integers divided by four is 189.5. Answer: Let x be the first even integer. Then, \(\frac{x+(x+2)}{4}\)=189.5.

Question 8. Subtract seven more than twice a number from the square of one-third of the number to get zero. Answer: Let x be the number. Then, (\(\frac{1}{3}\) x) 2 -(2x+7)=0.

Question 9. The sum of three consecutive integers is 42. Let x be the middle of the three integers. Transcribe the statement accordingly. Answer: (x-1)+x+(x+1)=42

Eureka Math Grade 8 Module 4 Lesson 1 Exit Ticket Answer Key

Question 1. When you square five times a number, you get three more than the number. Answer: Let x be the number. Then, (5x) 2 =x+3.

Question 2. Monica had some cookies. She gave seven to her sister. Then, she divided the remainder into two halves, and she still had five cookies left. Answer: Let x be the original amount of cookies. Then, \(\frac{1}{2}\) (x-7)=5.

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8.4: Chapter 8 Review Exercises

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Chapter Review Exercises

Distance and midpoint formulas; circles, exercise \(\pageindex{1}\) use the distance formula.

In the following exercises, find the distance between the points. Round to the nearest tenth if needed.

• \((-5,1)\) and \((-1,4)\)
• \((-2,5)\) and \((1,5)\)
• \((8,2)\) and \((-7,-3)\)
• \((1,-4)\) and \((5,-5)\)

4. \(d=\sqrt{17}, d \approx 4.1\)

Exercise \(\PageIndex{2}\) Use the Midpoint Formula

In the following exercises, find the midpoint of the line segments whose endpoints are given.

• \((-2,-6)\) and \((-4,-2)\)
• \((3,7)\) and \((5,1)\)
• \((-8,-10)\) and \((9,5)\)
• \((-3,2)\) and \((6,-9)\)

2. \((4,4)\)

4. \(\left(\frac{3}{2},-\frac{7}{2}\right)\)

Exercise \(\PageIndex{3}\) Write the Equation of a Circle in Standard Form

In the following exercises, write the standard form of the equation of the circle with the given information.

• radius is \(15\) and center is \((0,0)\)
• radius is \(\sqrt{7}\) and center is \((0,0)\)
• radius is \(9\) and center is \((-3,5)\)
• radius is \(7\) and center is \((-2,-5)\)
• center is \((3,6)\) and a point on the circle is \((3,-2)\)
• center is \((2,2)\) and a point on the circle is \((4,4)\)

2. \(x^{2}+y^{2}=7\)

4. \((x+2)^{2}+(y+5)^{2}=49\)

6. \((x-2)^{2}+(y-2)^{2}=8\)

Exercise \(\PageIndex{4}\) Graph a Circle

In the following exercises,

• Find the center and radius, then
• Graph each circle.
• \(2 x^{2}+2 y^{2}=450\)
• \(3 x^{2}+3 y^{2}=432\)
• \((x+3)^{2}+(y-5)^{2}=81\)
• \((x+2)^{2}+(y+5)^{2}=49\)
• \(x^{2}+y^{2}-6 x-12 y-19=0\)
• \(x^{2}+y^{2}-4 y-60=0\)
• radius: \(12,\) center: \((0,0)\)

• radius: \(7,\) center: \((-2,-5)\)

• radius: \(8,\) center: \((0,2)\)

Exercise \(\PageIndex{5}\) Graph Vertical Parabolas

In the following exercises, graph each equation by using its properties.

• \(y=x^{2}+4 x-3\)
• \(y=2 x^{2}+10 x+7\)
• \(y=-6 x^{2}+12 x-1\)
• \(y=-x^{2}+10 x\)

Exercise \(\PageIndex{6}\) Graph Vertical Parabolas

• Write the equation in standard form, then
• Use properties of the standard form to graph the equation.
• \(y=x^{2}+4 x+7\)
• \(y=2 x^{2}-4 x-2\)
• \(y=-3 x^{2}-18 x-29\)
• \(y=-x^{2}+12 x-35\)
• \(y=2(x-1)^{2}-4\)

• \(y=-(x-6)^{2}+1\)

Exercise \(\PageIndex{7}\) Graph Horizontal Parabolas

• \(x=2 y^{2}\)
• \(x=2 y^{2}+4 y+6\)
• \(x=-y^{2}+2 y-4\)
• \(x=-3 y^{2}\)

Exercise \(\PageIndex{8}\) Graph Horizontal Parabolas

• \(x=4 y^{2}+8 y\)
• \(x=y^{2}+4 y+5\)
• \(x=-y^{2}-6 y-7\)
• \(x=-2 y^{2}+4 y\)
• \(x=(y+2)^{2}+1\)

• \(x=-2(y-1)^{2}+2\)

Exercise \(\PageIndex{9}\) Solve Applications with Parabolas

In the following exercises, create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in standard form.

2. \(y=-\frac{1}{9} x^{2}+\frac{10}{3} x\)

Exercise \(\PageIndex{10}\) Graph an Ellipse with Center at the Origin

In the following exercises, graph each ellipse.

• \(\frac{x^{2}}{36}+\frac{y^{2}}{25}=1\)
• \(\frac{x^{2}}{4}+\frac{y^{2}}{81}=1\)
• \(49 x^{2}+64 y^{2}=3136\)
• \(9 x^{2}+y^{2}=9\)

Exercise \(\PageIndex{11}\) Find the Equation of an Ellipse with Center at the Origin

In the following exercises, find the equation of the ellipse shown in the graph.

2. \(\frac{x^{2}}{36}+\frac{y^{2}}{64}=1\)

Exercise \(\PageIndex{12}\) Graph an Ellipse with Center Not at the Origin

• \(\frac{(x-1)^{2}}{25}+\frac{(y-6)^{2}}{4}=1\)
• \(\frac{(x+4)^{2}}{16}+\frac{(y+1)^{2}}{9}=1\)
• \(\frac{(x-5)^{2}}{16}+\frac{(y+3)^{2}}{36}=1\)
• \(\frac{(x+3)^{2}}{9}+\frac{(y-2)^{2}}{25}=1\)

Exercise \(\PageIndex{13}\) Graph an Ellipse with Center Not at the Origin

• Write the equation in standard form and
• \(x^{2}+y^{2}+12 x+40 y+120=0\)
• \(25 x^{2}+4 y^{2}-150 x-56 y+321=0\)
• \(25 x^{2}+4 y^{2}+150 x+125=0\)
• \(4 x^{2}+9 y^{2}-126 x+405=0\)
• \(\frac{(x-3)^{2}}{4}+\frac{(y-7)^{2}}{25}=1\)

• \(\frac{x^{2}}{9}+\frac{(y-7)^{2}}{4}=1\)

Exercise \(\PageIndex{14}\) Solve Applications with Ellipses

In the following exercises, write the equation of the ellipse described.

• A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately \(10\) AU and the furthest is approximately \(90\) AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.

Exercise \(\PageIndex{15}\) Graph a Hyperbola with Center at \((0,0)\)

In the following exercises, graph.

• \(\frac{x^{2}}{25}-\frac{y^{2}}{9}=1\)
• \(\frac{y^{2}}{49}-\frac{x^{2}}{16}=1\)
• \(9 y^{2}-16 x^{2}=144\)
• \(16 x^{2}-4 y^{2}=64\)

Exercise \(\PageIndex{16}\) Graph a Hyperbola with Center at \((h,k)\)

• \(\frac{(x+1)^{2}}{4}-\frac{(y+1)^{2}}{9}=1\)
• \(\frac{(x-2)^{2}}{4}-\frac{(y-3)^{2}}{16}=1\)
• \(\frac{(y+2)^{2}}{9}-\frac{(x+1)^{2}}{9}=1\)
• \(\frac{(y-1)^{2}}{25}-\frac{(x-2)^{2}}{9}=1\)

Exercise \(\PageIndex{17}\) Graph a Hyperbola with Center at \((h,k)\)

• \(4 x^{2}-16 y^{2}+8 x+96 y-204=0\)
• \(16 x^{2}-4 y^{2}-64 x-24 y-36=0\)
• \(4 y^{2}-16 x^{2}+32 x-8 y-76=0\)
• \(36 y^{2}-16 x^{2}-96 x+216 y-396=0\)
• \(\frac{(x+1)^{2}}{16}-\frac{(y-3)^{2}}{4}=1\)

• \(\frac{(y-1)^{2}}{16}-\frac{(x-1)^{2}}{4}=1\)

Exercise \(\PageIndex{18}\) Identify the Graph of Each Equation as a Circle, Parabola, Ellipse, or Hyperbola

In the following exercises, identify the type of graph.

• \(16 y^{2}-9 x^{2}-36 x-96 y-36=0\)
• \(x^{2}+y^{2}-4 x+10 y-7=0\)
• \(y=x^{2}-2 x+3\)
• \(25 x^{2}+9 y^{2}=225\)
• \(x^{2}+y^{2}+4 x-10 y+25=0\)
• \(y^{2}-x^{2}-4 y+2 x-6=0\)
• \(x=-y^{2}-2 y+3\)
• \(16 x^{2}+9 y^{2}=144\)

Solve Systems of Nonlinear Equations

Exercise \(\pageindex{19}\) solve a system of nonlinear equations using graphing.

In the following exercises, solve the system of equations by using graphing.

• \(\left\{\begin{array}{l}{3 x^{2}-y=0} \\ {y=2 x-1}\end{array}\right.\)
• \(\left\{\begin{array}{l}{y=x^{2}-4} \\ {y=x-4}\end{array}\right.\)
• \(\left\{\begin{array}{l}{x^{2}+y^{2}=169} \\ {x=12}\end{array}\right.\)
• \(\left\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {y=-5}\end{array}\right.\)

Exercise \(\PageIndex{20}\) Solve a System of Nonlinear Equations Using Substitution

In the following exercises, solve the system of equations by using substitution.

• \(\left\{\begin{array}{l}{y=x^{2}+3} \\ {y=-2 x+2}\end{array}\right.\)
• \(\left\{\begin{array}{l}{x^{2}+y^{2}=4} \\ {x-y=4}\end{array}\right.\)
• \(\left\{\begin{array}{l}{9 x^{2}+4 y^{2}=36} \\ {y-x=5}\end{array}\right.\)
• \(\left\{\begin{array}{l}{x^{2}+4 y^{2}=4} \\ {2 x-y=1}\end{array}\right.\)

1. \((-1,4)\)

3. No solution

Exercise \(\PageIndex{21}\) Solve a System of Nonlinear Equations Using Elimination

In the following exercises, solve the system of equations by using elimination.

• \(\left\{\begin{array}{l}{x^{2}+y^{2}=16} \\ {x^{2}-2 y-1=0}\end{array}\right.\)
• \(\left\{\begin{array}{l}{x^{2}-y^{2}=5} \\ {-2 x^{2}-3 y^{2}=-30}\end{array}\right.\)
• \(\left\{\begin{array}{l}{4 x^{2}+9 y^{2}=36} \\ {3 y^{2}-4 x=12}\end{array}\right.\)
• \(\left\{\begin{array}{l}{x^{2}+y^{2}=14} \\ {x^{2}-y^{2}=16}\end{array}\right.\)

1. \((-\sqrt{7}, 3),(\sqrt{7}, 3)\)

3. \((-3,0),(0,-2),(0,2)\)

Exercise \(\PageIndex{22}\) Use a System of Nonlinear Equations to Solve Applications

In the following exercises, solve the problem using a system of equations.

• The sum of the squares of two numbers is \(25\). The difference of the numbers is \(1\). Find the numbers.
• The difference of the squares of two numbers is \(45\). The difference of the square of the first number and twice the square of the second number is \(9\). Find the numbers.
• The perimeter of a rectangle is \(58\) meters and its area is \(210\) square meters. Find the length and width of the rectangle.
• Colton purchased a larger microwave for his kitchen. The diagonal of the front of the microwave measures \(34\) inches. The front also has an area of \(480\) square inches. What are the length and width of the microwave?

1. \(-3\) and \(-4\) or \(4\) and \(3\)

3. If the length is \(14\) inches, the width is \(15\) inches. If the length is \(15\) inches, the width is \(14\) inches.

Practice Test

Exercise \(\pageindex{23}\).

In the following exercises, find the distance between the points and the midpoint of the line segment with the given endpoints. Round to the nearest tenth as needed.

• \((-4,-3)\) and \((-10,-11)\)
• \((6,8)\) and \((-5,-3)\)

1. distance: \(10,\) midpoint: \((-7,-7)\)

Exercise \(\PageIndex{24}\)

• radius is \(11\) and center is \((0,0)\)
• radius is \(12\) and center is \((10,-2)\)
• center is \((-2,3)\) and a point on the circle is \((2,-3)\)
• Find the equation of the ellipse shown in the graph.

1. \(x^{2}+y^{2}=121\)

3. \((x+2)^{2}+(y-3)^{2}=52\)

Exercise \(\PageIndex{25}\)

• Identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, and
• Graph the equation.
• \(4 x^{2}+49 y^{2}=196\)
• \(y=3(x-2)^{2}-2\)
• \(3 x^{2}+3 y^{2}=27\)
• \(\frac{y^{2}}{100}-\frac{x^{2}}{36}=1\)
• \(\frac{x^{2}}{16}+\frac{y^{2}}{81}=1\)
• \(x=2 y^{2}+10 y+7\)
• \(64 x^{2}-9 y^{2}=576\)

Exercise \(\PageIndex{26}\)

• Identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola,
• Write the equation in standard form, and
• \(25 x^{2}+64 y^{2}+200 x-256 y-944=0\)
• \(x^{2}+y^{2}+10 x+6 y+30=0\)
• \(9 x^{2}-25 y^{2}-36 x-50 y-214=0\)
• \(y=x^{2}+6 x+8\)
• Solve the nonlinear system of equations by graphing: \(\left\{\begin{array}{l}{3 y^{2}-x=0} \\ {y=-2 x-1}\end{array}\right.\).
• Solve the nonlinear system of equations using substitution: \(\left\{\begin{array}{l}{x^{2}+y^{2}=8} \\ {y=-x-4}\end{array}\right.\).
• Solve the nonlinear system of equations using elimination: \(\left\{\begin{array}{l}{x^{2}+9 y^{2}=9} \\ {2 x^{2}-9 y^{2}=18}\end{array}\right.\)
• Create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in \(y=a x^{2}+b x+c\) form.

10. A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately \(20\) AU and the furthest is approximately \(70\) AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.

11. The sum of two numbers is \(22\) and the product is \(−240\). Find the numbers.

12. For her birthday, Olive’s grandparents bought her a new widescreen TV. Before opening it she wants to make sure it will fit her entertainment center. The TV is \(55\)”. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of \(1452\) square inches. Her entertainment center has an insert for the TV with a length of \(50\) inches and width of \(40\) inches. What are the length and width of the TV screen and will it fit Olive’s entertainment center?

• \((x+5)^{2}+(y+3)^{2}=4\)

• \(\frac{(x-2)^{2}}{25}-\frac{(y+1)^{2}}{9}=1\)

6. No solution

8. \((0,-3),(0,3)\)

10. \(\frac{x^{2}}{2025}+\frac{y^{2}}{1400}=1\)

12. The length is \(44\) inches and the width is \(33\) inches. The TV will fit Olive’s entertainment center.

Home > CC1 > Chapter Ch8 > Lesson 8.1.4

Lesson 8.1.1, lesson 8.1.2, lesson 8.1.3, lesson 8.1.4, lesson 8.1.5, lesson 8.2.1, lesson 8.3.1, lesson 8.3.2, lesson 8.3.3.

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