Systems by Graphing

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Answer Key & Explanations

Systems by Graphing · Grade 8 · Worksheet 1

1. Emma is graphing two linear equations on the same coordinate plane. The first equation is y = -2x + 10, and the second equation is y = (1/2)x - 5. What is the intersection point of these two lines? Answer: (6, -2) Solution: Set the two equations equal to each other since at the intersection, the y-values are the same: -2x + 10 = (1/2)x - 5 Solve for x.
   Full step-by-step solution

   Step 1: Set the two equations equal to each other since at the intersection, the y-values are the same: -2x + 10 = (1/2)x - 5 Step 2: Solve for x. Multiply both sides by 2 to eliminate the fraction: 2(-2x + 10) = 2((1/2)x - 5) -4x + 20 = x - 10 Step 3: Add 4x to both sides: 20 = 5x - 10 Step 4: Add 10 to both sides: 30 = 5x Step 5: Divide both sides by 5: x = 6 Step 6: Substitute x = 6 into the first equation to find y: y = -2(6) + 10 = -12 + 10 = -2 Step 7: Verify by substituting into the second equation: y = (1/2)(6) - 5 = 3 - 5 = -2 Both equations give y = -2, so the intersection point is (6, -2). The answer is (6, -2).

2. A scientist is studying two bacterial cultures. Culture A starts with 800 bacteria and grows at a rate of 50 bacteria per hour. Culture B starts with 200 bacteria and grows at a rate of 100 bacteria per hour. After how many hours will both cultures have the same number of bacteria? Answer: 12 Solution: Culture A starts with 800 bacteria and grows at 50 bacteria per hour. So after h hours, the number of bacteria in Culture A is: A = 800 + 50h Culture B starts with 200 bacteria and grows at 100 bacteria per hour.
   Full step-by-step solution

   Let's define the number of hours as h. Culture A starts with 800 bacteria and grows at 50 bacteria per hour. So after h hours, the number of bacteria in Culture A is: A = 800 + 50h Culture B starts with 200 bacteria and grows at 100 bacteria per hour. So after h hours, the number of bacteria in Culture B is: B = 200 + 100h We want to find h when both cultures have the same number of bacteria, so set A equal to B: 800 + 50h = 200 + 100h Now solve for h step by step. First, subtract 50h from both sides: 800 = 200 + 100h - 50h 800 = 200 + 50h Next, subtract 200 from both sides: 800 - 200 = 50h 600 = 50h Finally, divide both sides by 50: 600 / 50 = h h = 12 So after 12 hours, both cultures will have the same number of bacteria. Let's check: Culture A: 800 + 50*12 = 800 + 600 = 1400 Culture B: 200 + 100*12 = 200 + 1200 = 1400 Yes, both are equal. Answer: 12 hours

3. y = 4x - 7 and y = -2x + 11 Answer: (3, 5) Solution: Step 1: Set the equations equal to find x: 4x - 7 = -2x + 11 Step 2: Add 2x to both sides: 6x - 7 = 11 Step 3: Add 7 to both sides: 6x = 18 Step 4: Divide both sides by 6: x = 3 Step 5: Substitute x = 3 into y = 4x - 7: y = 4(3) - 7 = 12 - 7 = 5 Step 6: Verify by substituting into the other…
   Full step-by-step solution

   Step 1: Set the equations equal to find x: 4x - 7 = -2x + 11 Step 2: Add 2x to both sides: 6x - 7 = 11 Step 3: Add 7 to both sides: 6x = 18 Step 4: Divide both sides by 6: x = 3 Step 5: Substitute x = 3 into y = 4x - 7: y = 4(3) - 7 = 12 - 7 = 5 Step 6: Verify by substituting into the other equation: y = -2(3) + 11 = -6 + 11 = 5 ✓ Step 7: The solution is the intersection point: (3, 5)

4. 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.

5. Hana is helping her uncle at his weekend market stall. He sells two types of fruit bundles: a small bundle of apples for $7 each and a large bundle of mixed fruit for $12 each. At the end of the day, Hana counts that they sold a total of 28 bundles altogether, and the total money collected was $261. Hana wants to graph the system of equations representing this situation to find exactly how many small bundles (x) and how many large bundles (y) were sold. After graphing the two equations, what is the point of intersection? Answer: (15, 13) Solution: Define the variables. Let x = number of small bundles, y = number of large bundles. Write the equation for the total number of bundles: x + y = 28.
   Full step-by-step solution

   Step 1: Define the variables. Let x = number of small bundles, y = number of large bundles. Step 2: Write the equation for the total number of bundles: x + y = 28. Step 3: Write the equation for the total money collected: 7x + 12y = 261. Step 4: Solve the system by graphing. First, rewrite each equation in slope-intercept form (y = mx + b). For x + y = 28, subtract x from both sides: y = -x + 28. For 7x + 12y = 261, subtract 7x: 12y = -7x + 261, then divide by 12: y = (-7/12)x + 21.75. Step 5: Find the intersection by setting the equations equal: -x + 28 = (-7/12)x + 21.75. Step 6: Multiply both sides by 12 to eliminate the fraction: -12x + 336 = -7x + 261. Step 7: Add 12x to both sides: 336 = 5x + 261. Subtract 261: 75 = 5x. Divide by 5: x = 15. Step 8: Substitute x = 15 into y = -x + 28: y = -15 + 28 = 13. Step 9: Verify with the money equation: 7(15) + 12(13) = 105 + 156 = 261. Correct. The point of intersection is (15, 13).

6. Mere graphs the lines y = 2x - 3 and y = -x + 12 on the same coordinate plane. What is the intersection point of these two lines? Answer: (5, 7) Solution: The intersection point is where the y-values of both equations are equal. Set the right sides equal: 2x - 3 = -x + 12. Add x to both sides: 2x + x - 3 = 12, so 3x - 3 = 12.
   Full step-by-step solution

   Step 1: The intersection point is where the y-values of both equations are equal. Set the right sides equal: 2x - 3 = -x + 12. Step 2: Add x to both sides: 2x + x - 3 = 12, so 3x - 3 = 12. Step 3: Add 3 to both sides: 3x = 15. Step 4: Divide by 3: x = 5. Step 5: Substitute x = 5 into y = 2x - 3: y = 2(5) - 3 = 10 - 3 = 7. Step 6: Check with the second equation: y = -5 + 12 = 7. Both match. Step 7: The intersection point is (5, 7). The answer is (5, 7).

7. Kaia graphs two lines on a coordinate plane. The first line passes through the points (1, 3) and (5, 7). The second line passes through the points (1, 7) and (5, 3). What is the intersection point of these two lines? Answer: (4, 6) Solution: Find the equation of the first line through (1, 3) and (5, 7). Slope m = (7 - 3)/(5 - 1) = 4/4 = 1. Using point-slope form: y - 3 = 1(x - 1) => y = x + 2.
   Full step-by-step solution

   Step 1: Find the equation of the first line through (1, 3) and (5, 7). Slope m = (7 - 3)/(5 - 1) = 4/4 = 1. Using point-slope form: y - 3 = 1(x - 1) => y = x + 2. Step 2: Find the equation of the second line through (1, 7) and (5, 3). Slope m = (3 - 7)/(5 - 1) = (-4)/4 = -1. Using point-slope form: y - 7 = -1(x - 1) => y = -x + 8. Step 3: Set the equations equal to find the intersection: x + 2 = -x + 8 => 2x = 6 => x = 3. Step 4: Substitute x = 3 into y = x + 2: y = 3 + 2 = 5. The intersection point is (3, 5). Verification: Check in second equation: y = -3 + 8 = 5. Correct. The answer is (3, 5).