Worksheet 1Worksheet 2Worksheet 3

The download includes this worksheet followed by the answer key with explanations. In the print dialog choose “Save as PDF” to download.

lessonbunny.com
Name: ______________________________ Date: ______________

Systems by Graphing

Grade 8 · Algebra · Worksheet 2

  1. 2x + y = 10 and y = 3x - 5 Answer: ______________
  2. y = 3x - 5 and y = -2x + 8 Answer: ______________
  3. y = 4x - 9 and y = -2x + 3 Answer: ______________
  4. Liam graphs the lines y = 3x - 7 and y = -x + 9 on the same coordinate grid. What are the coordinates of the point where the two lines intersect? Answer: ______________
  5. 3x + 2y = 12 and y = -x + 5 Answer: ______________
  6. 2x + y = 8 and y = -x + 5 Answer: ______________
  7. Olivia is planning a school field day and needs to choose between two companies for renting inflatable bounce houses. BounceFun charges a $45 setup fee plus $9 per hour. JumpZone charges a $15 setup fee plus $12 per hour. Olivia wants to graph both cost equations to determine after how many hours the total cost for both companies would be the same. Let x represent the number of hours and y represent the total cost in dollars. After graphing the system of equations, what is the point of intersection (x, y)? Answer: ______________
  8. Liam is planning a school fundraiser and needs to decide between selling cupcakes or cookies. The cupcake stand costs $15 to set up and makes a profit of $2 per cupcake sold. The cookie stand costs $10 to set up and makes a profit of $1.50 per cookie sold. Liam wants to know how many items he would need to sell for both stands to make the same total profit. Write your answer as an ordered pair (x,y) where x represents the number of cupcakes and y represents the number of cookies. Answer: ______________
lessonbunny.com

Answer Key & Explanations

Systems by Graphing · Grade 8 · Worksheet 2

  1. 2x + y = 10 and y = 3x - 5 Answer: (3, 4) Solution: The first equation is 2x + y = 10. Rewrite it in slope-intercept form: y = -2x + 10. The second equation is already in slope-intercept form: y = 3x - 5.
    Full step-by-step solution

    Step 1: The first equation is 2x + y = 10. Rewrite it in slope-intercept form: y = -2x + 10. Step 2: The second equation is already in slope-intercept form: y = 3x - 5. Step 3: Graph y = -2x + 10. The y-intercept is 10, and the slope is -2 (down 2, right 1). Step 4: Graph y = 3x - 5. The y-intercept is -5, and the slope is 3 (up 3, right 1). Step 5: The two lines intersect at the point (3, 4). Step 6: Verify by substituting x = 3 into both equations: For 2x + y = 10: 2(3) + 4 = 6 + 4 = 10 ✓ For y = 3x - 5: 4 = 3(3) - 5 = 9 - 5 = 4 ✓ The solution is (3, 4).

  2. y = 3x - 5 and y = -2x + 8 Answer: (2.6, 2.8) Solution: Step 1: Set the equations equal to find x: 3x - 5 = -2x + 8 Step 2: Add 2x to both sides: 5x - 5 = 8 Step 3: Add 5 to both sides: 5x = 13 Step 4: Divide both sides by 5: x = 13/5 = 2.6 Step 5: Substitute x = 2.6 into y = 3x - 5: y = 3(2.6) - 5 = 7.8 - 5 = 2.8 Step 6: Verify by substituting into…
    Full step-by-step solution

    Step 1: Set the equations equal to find x: 3x - 5 = -2x + 8 Step 2: Add 2x to both sides: 5x - 5 = 8 Step 3: Add 5 to both sides: 5x = 13 Step 4: Divide both sides by 5: x = 13/5 = 2.6 Step 5: Substitute x = 2.6 into y = 3x - 5: y = 3(2.6) - 5 = 7.8 - 5 = 2.8 Step 6: Verify by substituting into the other equation: y = -2(2.6) + 8 = -5.2 + 8 = 2.8 ✓ Step 7: The solution is the intersection point: (2.6, 2.8)

  3. y = 4x - 9 and y = -2x + 3 Answer: (2, -1) Solution: Step 1: Set the equations equal to find x: 4x - 9 = -2x + 3 Step 2: Add 2x to both sides: 6x - 9 = 3 Step 3: Add 9 to both sides: 6x = 12 Step 4: Divide both sides by 6: x = 2 Step 5: Substitute x = 2 into y = 4x - 9: y = 4(2) - 9 = 8 - 9 = -1 Step 6: Verify by substituting into the other…
    Full step-by-step solution

    Step 1: Set the equations equal to find x: 4x - 9 = -2x + 3 Step 2: Add 2x to both sides: 6x - 9 = 3 Step 3: Add 9 to both sides: 6x = 12 Step 4: Divide both sides by 6: x = 2 Step 5: Substitute x = 2 into y = 4x - 9: y = 4(2) - 9 = 8 - 9 = -1 Step 6: Verify by substituting into the other equation: y = -2(2) + 3 = -4 + 3 = -1 ✓ Step 7: The solution is the ordered pair (2, -1) The answer is (2, -1).

  4. Liam graphs the lines y = 3x - 7 and y = -x + 9 on the same coordinate grid. What are the coordinates of the point where the two lines intersect? Answer: (4, 5) Solution: To find the intersection point, set the two equations equal to each other because at the intersection, the y-values are the same. 3x - 7 = -x + 9 Solve for x.
    Full step-by-step solution

    Step 1: To find the intersection point, set the two equations equal to each other because at the intersection, the y-values are the same. 3x - 7 = -x + 9 Step 2: Solve for x. 3x + x = 9 + 7 4x = 16 x = 4 Step 3: Substitute x = 4 into either original equation to find y. Using y = 3x - 7: y = 3(4) - 7 y = 12 - 7 y = 5 Step 4: Verify by substituting x = 4 into the other equation, y = -x + 9: y = -4 + 9 = 5 ✓ Step 5: The intersection point is (4, 5). Final answer: (4, 5)

  5. 3x + 2y = 12 and y = -x + 5 Answer: (2, 3) Solution: Substitute y = -x + 5 into the first equation: 3x + 2(-x + 5) = 12 Distribute the 2: 3x - 2x + 10 = 12 Combine like terms: x + 10 = 12 Subtract 10 from both sides: x = 2 Substitute x = 2 into y = -x + 5: y = -2 + 5 = 3 The solution is the ordered pair (2, 3).
    Full step-by-step solution

    Step 1: Substitute y = -x + 5 into the first equation: 3x + 2(-x + 5) = 12 Step 2: Distribute the 2: 3x - 2x + 10 = 12 Step 3: Combine like terms: x + 10 = 12 Step 4: Subtract 10 from both sides: x = 2 Step 5: Substitute x = 2 into y = -x + 5: y = -2 + 5 = 3 Step 6: The solution is the ordered pair (2, 3).

  6. 2x + y = 8 and y = -x + 5 Answer: (1, 6) Solution: Substitute the expression for y from the second equation into the first equation. 2x + (-x + 5) = 8 Simplify and solve for x.
    Full step-by-step solution

    Step 1: Substitute the expression for y from the second equation into the first equation. 2x + (-x + 5) = 8 Step 2: Simplify and solve for x. 2x - x + 5 = 8 x + 5 = 8 x = 8 - 5 x = 3 Step 3: Substitute x = 3 back into the second equation to find y. y = -3 + 5 y = 2 Step 4: The solution is the ordered pair (x, y). The solution is (3, 2).

  7. Olivia is planning a school field day and needs to choose between two companies for renting inflatable bounce houses. BounceFun charges a $45 setup fee plus $9 per hour. JumpZone charges a $15 setup fee plus $12 per hour. Olivia wants to graph both cost equations to determine after how many hours the total cost for both companies would be the same. Let x represent the number of hours and y represent the total cost in dollars. After graphing the system of equations, what is the point of intersection (x, y)? Answer: (10, 135) Solution: Write the system of equations. BounceFun: y = 9x + 45. JumpZone: y = 12x + 15.
    Full step-by-step solution

    Step 1: Write the system of equations. BounceFun: y = 9x + 45. JumpZone: y = 12x + 15. Step 2: Set the equations equal to find the x-coordinate of the intersection: 9x + 45 = 12x + 15. Step 3: Solve for x. Subtract 9x from both sides: 45 = 3x + 15. Subtract 15 from both sides: 30 = 3x. Divide both sides by 3: x = 10. Step 4: Substitute x = 10 into either equation to find y. Using y = 9x + 45: y = 9(10) + 45 = 90 + 45 = 135. Step 5: The point of intersection is (10, 135). This means after 10 hours, both companies charge $135.

  8. Liam is planning a school fundraiser and needs to decide between selling cupcakes or cookies. The cupcake stand costs $15 to set up and makes a profit of $2 per cupcake sold. The cookie stand costs $10 to set up and makes a profit of $1.50 per cookie sold. Liam wants to know how many items he would need to sell for both stands to make the same total profit. Write your answer as an ordered pair (x,y) where x represents the number of cupcakes and y represents the number of cookies. Answer: (10,20) Solution: In business decisions, we often compare different options using systems of equations. Each option can be represented by a linear equation where the total value depends on a fixed cost and a variable cost per unit.
    Full step-by-step solution

    In business decisions, we often compare different options using systems of equations. Each option can be represented by a linear equation where the total value depends on a fixed cost and a variable cost per unit. The point where two lines intersect represents the situation where both options yield the same result, which helps in making informed choices between alternatives.