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Systems by Graphing

Grade 8 · Algebra · Worksheet 2

  1. y = 5x - 8 and y = -2x + 6 Answer: ______________
  2. Emma is planning a school trip to the science museum and needs to choose between two transportation companies. Speedy Buses charges a $200 flat fee plus $8 per student. Reliable Coaches charges a $50 flat fee plus $12 per student. Emma wants to graph both cost equations to determine for how many students the total cost would be the same with both companies. Let x represent the number of students and y represent the total cost in dollars. After graphing the system of equations, what is the point of intersection? Answer: ______________
  3. Liam is planning a school fundraiser selling t-shirts and hoodies. The t-shirts cost $8 each and the hoodies cost $15 each. He needs to raise exactly $500. If he sells 40 items in total, how many of each type did he sell? Answer: ______________
  4. 2x + 3y = 12 and y = -x + 4 Answer: ______________
  5. Emma is planning a school trip to the science museum and needs to decide between two bus companies. Speedy Buses charges a $75 flat fee plus $2 per student, while Reliable Rides charges a $25 flat fee plus $4 per student. Emma wants to graph both cost equations to determine how many students would make both companies charge the same total amount. Write your answer as an ordered pair (number of students, total cost). Answer: ______________
  6. Matiu is helping his school's kapa haka group plan a fundraiser by selling raffle tickets. They are considering two different pricing options. Option A has a fixed booth rental fee of $60 plus $2 per ticket sold. Option B has a fixed booth rental fee of $30 plus $4 per ticket sold. Matiu wants to graph both cost equations to find out how many tickets they would need to sell for both options to have the same total cost. Let x represent the number of tickets sold and y represent the total cost in dollars. After graphing the system of equations, what is the point of intersection? Answer: ______________
  7. y = 3x - 7 and y = -2x + 8 Answer: ______________
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Answer Key & Explanations

Systems by Graphing · Grade 8 · Worksheet 2

  1. y = 5x - 8 and y = -2x + 6 Answer: (2, 2) Solution: Step 1: Set the equations equal to find x: 5x - 8 = -2x + 6 Step 2: Add 2x to both sides: 7x - 8 = 6 Step 3: Add 8 to both sides: 7x = 14 Step 4: Divide both sides by 7: x = 2 Step 5: Substitute x = 2 into y = 5x - 8: y = 5(2) - 8 = 10 - 8 = 2 Step 6: Verify by substituting into the other…
    Full step-by-step solution

    Step 1: Set the equations equal to find x: 5x - 8 = -2x + 6 Step 2: Add 2x to both sides: 7x - 8 = 6 Step 3: Add 8 to both sides: 7x = 14 Step 4: Divide both sides by 7: x = 2 Step 5: Substitute x = 2 into y = 5x - 8: y = 5(2) - 8 = 10 - 8 = 2 Step 6: Verify by substituting into the other equation: y = -2(2) + 6 = -4 + 6 = 2 ✓ Step 7: The solution is the ordered pair (2, 2) The answer is (2, 2).

  2. Emma is planning a school trip to the science museum and needs to choose between two transportation companies. Speedy Buses charges a $200 flat fee plus $8 per student. Reliable Coaches charges a $50 flat fee plus $12 per student. Emma wants to graph both cost equations to determine for how many students the total cost would be the same with both companies. Let x represent the number of students and y represent the total cost in dollars. After graphing the system of equations, what is the point of intersection? Answer: (37.5, 500) Solution: Write the system of equations. For Speedy Buses: y = 8x + 200 For Reliable Coaches: y = 12x + 50 Set the equations equal to find where they intersect. 8x + 200 = 12x + 50 Solve for x.
    Full step-by-step solution

    Step 1: Write the system of equations. For Speedy Buses: y = 8x + 200 For Reliable Coaches: y = 12x + 50 Step 2: Set the equations equal to find where they intersect. 8x + 200 = 12x + 50 Step 3: Solve for x. 200 - 50 = 12x - 8x 150 = 4x x = 150/4 x = 37.5 Step 4: Substitute x = 37.5 into either equation to find y. Using y = 8x + 200: y = 8(37.5) + 200 y = 300 + 200 y = 500 Step 5: Write the solution as an ordered pair. The point of intersection is (37.5, 500). This means at 37.5 students, both companies charge $500.

  3. Liam is planning a school fundraiser selling t-shirts and hoodies. The t-shirts cost $8 each and the hoodies cost $15 each. He needs to raise exactly $500. If he sells 40 items in total, how many of each type did he sell? Answer: 20 t-shirts and 20 hoodies Solution: When solving systems of equations with two variables, you can represent the situation with two equations.
    Full step-by-step solution

    When solving systems of equations with two variables, you can represent the situation with two equations. One equation typically relates to quantities (like total items) while the other relates to values (like total cost). Graphing both equations on a coordinate plane shows where they intersect, which represents the solution that satisfies both conditions simultaneously.

  4. 2x + 3y = 12 and y = -x + 4 Answer: (0, 4) Solution: 1) 2x + 3y = 12 2) y = -x + 4 Substitute equation (2) into equation (1). Since y = -x + 4, we replace y in equation (1) with (-x + 4): 2x + 3(-x + 4) = 12 Simplify and solve for x.
    Full step-by-step solution

    We are given the system of equations: 1) 2x + 3y = 12 2) y = -x + 4 Step 1: Substitute equation (2) into equation (1). Since y = -x + 4, we replace y in equation (1) with (-x + 4): 2x + 3(-x + 4) = 12 Step 2: Simplify and solve for x. 2x - 3x + 12 = 12 (2x - 3x) = -x, so: -x + 12 = 12 Step 3: Isolate x. Subtract 12 from both sides: -x + 12 - 12 = 12 - 12 -x = 0 Multiply both sides by -1: x = 0 Step 4: Substitute x = 0 into equation (2) to find y. y = -0 + 4 y = 4 Step 5: Write the solution as an ordered pair. (x, y) = (0, 4) Step 6: Check in equation (1). 2(0) + 3(4) = 0 + 12 = 12, correct. Final answer: (0, 4)

  5. Emma is planning a school trip to the science museum and needs to decide between two bus companies. Speedy Buses charges a $75 flat fee plus $2 per student, while Reliable Rides charges a $25 flat fee plus $4 per student. Emma wants to graph both cost equations to determine how many students would make both companies charge the same total amount. Write your answer as an ordered pair (number of students, total cost). Answer: (25, 125) Solution: Speedy Buses: y = 2x + 75 Reliable Rides: y = 4x + 25 Set the equations equal to find where they intersect 2x + 75 = 4x + 25 75 - 25 = 4x - 2x 50 = 2x x = 25 Substitute x = 25 into either equation to find y y = 2(25) + 75 y = 50 + 75 y = 125 (25, 125) This means with 25 students, both companies…
    Full step-by-step solution

    Step 1: Write the equations for both companies Speedy Buses: y = 2x + 75 Reliable Rides: y = 4x + 25 Step 2: Set the equations equal to find where they intersect 2x + 75 = 4x + 25 Step 3: Solve for x 75 - 25 = 4x - 2x 50 = 2x x = 25 Step 4: Substitute x = 25 into either equation to find y y = 2(25) + 75 y = 50 + 75 y = 125 Step 5: Write the answer as an ordered pair (25, 125) This means with 25 students, both companies charge $125.

  6. Matiu is helping his school's kapa haka group plan a fundraiser by selling raffle tickets. They are considering two different pricing options. Option A has a fixed booth rental fee of $60 plus $2 per ticket sold. Option B has a fixed booth rental fee of $30 plus $4 per ticket sold. Matiu wants to graph both cost equations to find out how many tickets they would need to sell for both options to have the same total cost. Let x represent the number of tickets sold and y represent the total cost in dollars. After graphing the system of equations, what is the point of intersection? Answer: (15, 90) Solution: Write the system of equations. For Option A: y = 2x + 60. For Option B: y = 4x + 30.
    Full step-by-step solution

    Step 1: Write the system of equations. For Option A: y = 2x + 60. For Option B: y = 4x + 30. Step 2: Set the equations equal to each other to find the intersection point: 2x + 60 = 4x + 30. Step 3: Solve for x. Subtract 2x from both sides: 60 = 2x + 30. Subtract 30 from both sides: 30 = 2x. Divide both sides by 2: x = 15. Step 4: Substitute x = 15 into either equation to find y. Using y = 2x + 60: y = 2(15) + 60 = 30 + 60 = 90. Step 5: Write the solution as an ordered pair: (15, 90). This means at 15 tickets sold, both options cost $90.

  7. y = 3x - 7 and y = -2x + 8 Answer: (3, 2) Solution: Step 1: Set the equations equal to find x: 3x - 7 = -2x + 8 Step 2: Add 2x to both sides: 5x - 7 = 8 Step 3: Add 7 to both sides: 5x = 15 Step 4: Divide both sides by 5: x = 3 Step 5: Substitute x = 3 into y = 3x - 7: y = 3(3) - 7 = 9 - 7 = 2 Step 6: Verify by substituting into the other…
    Full step-by-step solution

    Step 1: Set the equations equal to find x: 3x - 7 = -2x + 8 Step 2: Add 2x to both sides: 5x - 7 = 8 Step 3: Add 7 to both sides: 5x = 15 Step 4: Divide both sides by 5: x = 3 Step 5: Substitute x = 3 into y = 3x - 7: y = 3(3) - 7 = 9 - 7 = 2 Step 6: Verify by substituting into the other equation: y = -2(3) + 8 = -6 + 8 = 2 ✓ Step 7: The solution is the ordered pair (3, 2) The answer is (3, 2).