Systems by Graphing

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Answer Key & Explanations

Systems by Graphing · Grade 8 · Worksheet 3

1. y = 3x - 4 and y = -2x + 6 Answer: (2, 2) Solution: Set the equations equal to find the x-coordinate of intersection: 3x - 4 = -2x + 6 Add 2x to both sides: 5x - 4 = 6 Add 4 to both sides: 5x = 10 Divide both sides by 5: x = 2 Substitute x = 2 into either original equation to find y: y = 3(2) - 4 = 6 - 4 = 2 The solution is the ordered pair (2,…
   Full step-by-step solution

   Step 1: Set the equations equal to find the x-coordinate of intersection: 3x - 4 = -2x + 6 Step 2: Add 2x to both sides: 5x - 4 = 6 Step 3: Add 4 to both sides: 5x = 10 Step 4: Divide both sides by 5: x = 2 Step 5: Substitute x = 2 into either original equation to find y: y = 3(2) - 4 = 6 - 4 = 2 Step 6: The solution is the ordered pair (2, 2), which represents the point where the two lines intersect. The answer is (2, 2).

2. 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).

3. Liam is comparing two options for renting a bike for the weekend. Bike Shop A charges a flat fee of $25 plus $10 per hour. Bike Shop B charges a flat fee of $15 plus $15 per hour. Liam wants to graph both cost equations to find out after how many hours the total cost from both shops will be the same. Let x represent the number of hours and y represent the total cost in dollars. What is the point of intersection (x, y) where the two lines cross? Answer: (2, 45) Solution: Write the equations. For Shop A: y = 10x + 25. For Shop B: y = 15x + 15.
   Full step-by-step solution

   Step 1: Write the equations. For Shop A: y = 10x + 25. For Shop B: y = 15x + 15. Step 2: Set the equations equal to each other to find the x-coordinate of the intersection: 10x + 25 = 15x + 15. Step 3: Solve for x. Subtract 10x from both sides: 25 = 5x + 15. Subtract 15 from both sides: 10 = 5x. Divide by 5: x = 2. Step 4: Substitute x = 2 into either equation to find y. Using Shop A: y = 10(2) + 25 = 20 + 25 = 45. Using Shop B: y = 15(2) + 15 = 30 + 15 = 45. Step 5: The point of intersection is (2, 45). This means after 2 hours, both shops charge $45.

4. Solve the system by graphing: y = 3x - 7 and y = -x + 9 Answer: (4, 5) Solution: Step 1: Set the equations equal to find x: 3x - 7 = -x + 9 Step 2: Add x to both sides: 4x - 7 = 9 Step 3: Add 7 to both sides: 4x = 16 Step 4: Divide both sides by 4: x = 4 Step 5: Substitute x = 4 into y = 3x - 7: y = 3(4) - 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: 3x - 7 = -x + 9 Step 2: Add x to both sides: 4x - 7 = 9 Step 3: Add 7 to both sides: 4x = 16 Step 4: Divide both sides by 4: x = 4 Step 5: Substitute x = 4 into y = 3x - 7: y = 3(4) - 7 = 12 - 7 = 5 Step 6: Verify by substituting into the other equation: y = -4 + 9 = 5 ✓ Step 7: The solution is the intersection point: (4, 5)

5. Charlotte is comparing two summer job offers. At the local bookstore, she earns a base pay of $30 per day plus $9 per hour worked. At the coffee shop, she earns a base pay of $10 per day plus $12 per hour worked. Charlotte wants to graph both earnings equations to determine after how many hours of work her total daily pay would be the same at both jobs. Let x represent the number of hours worked in a day and y represent the total daily pay in dollars. After graphing the system of equations, what is the point of intersection? Answer: (6.67, 90) Solution: Write the system of equations. Bookstore: y = 9x + 30 Coffee shop: y = 12x + 10 Set the equations equal to find the x-coordinate of the intersection. 9x + 30 = 12x + 10 Solve for x.
   Full step-by-step solution

   Step 1: Write the system of equations. Bookstore: y = 9x + 30 Coffee shop: y = 12x + 10 Step 2: Set the equations equal to find the x-coordinate of the intersection. 9x + 30 = 12x + 10 Step 3: Solve for x. 30 - 10 = 12x - 9x 20 = 3x x = 20/3 x = 6.67 (rounded to two decimal places) Step 4: Substitute x = 20/3 into either equation to find y. Using y = 9x + 30: y = 9(20/3) + 30 y = 60 + 30 y = 90 Step 5: Write the solution as an ordered pair. The point of intersection is (20/3, 90) or approximately (6.67, 90). This means that after about 6.67 hours, both jobs pay $90 for the day.

6. A rectangular garden has a length that is 5 meters more than twice its width. The perimeter of the garden is 46 meters. On a coordinate plane, if you graph the equations for length (l = 2w + 5) and perimeter (2l + 2w = 46), at what point (w, l) do the two lines intersect? Answer: (6, 17) Solution: Write down the given equations. 1. Length is 5 more than twice the width: \( l = 2w + 5 \) 2.
   Full step-by-step solution

   Let's solve this step-by-step. --- **Step 1: Write down the given equations.** From the problem: 1. Length is 5 more than twice the width: \( l = 2w + 5 \) 2. Perimeter is 46 meters: \( 2l + 2w = 46 \) --- **Step 2: Simplify the perimeter equation.** Divide both sides of \( 2l + 2w = 46 \) by 2: \( l + w = 23 \) So now we have the system: \( l = 2w + 5 \) \( l + w = 23 \) --- **Step 3: Substitute \( l \) from the first equation into the second equation.** From \( l = 2w + 5 \), substitute into \( l + w = 23 \): \( (2w + 5) + w = 23 \) --- **Step 4: Combine like terms and solve for \( w \).** \( 2w + 5 + w = 23 \) \( 3w + 5 = 23 \) \( 3w = 23 - 5 \) \( 3w = 18 \) \( w = 6 \) --- **Step 5: Find \( l \) using \( w = 6 \).** \( l = 2w + 5 \) \( l = 2(6) + 5 \) \( l = 12 + 5 \) \( l = 17 \) --- **Step 6: Interpret the result in terms of the graph.** The two lines \( l = 2w + 5 \) and \( 2l + 2w = 46 \) (or \( l + w = 23 \)) intersect at the point \( (w, l) = (6, 17) \). --- **Final Answer:** (6, 17)

7. Noah is organizing a school bake sale and comparing two options for buying ingredients. The local grocery store charges a $35 delivery fee plus $7 per batch of cookies. The wholesale market charges a $15 delivery fee plus $9 per batch of cookies. Noah wants to graph the total cost equations for both options to find out how many batches of cookies would make the total cost the same. Let x represent the number of batches and y represent the total cost in dollars. After graphing the system of equations, what is the point of intersection? Answer: (10, 105) Solution: Write the system of equations. Grocery store: y = 7x + 35 Wholesale market: y = 9x + 15 Set the equations equal to find the intersection. 7x + 35 = 9x + 15 Solve for x.
   Full step-by-step solution

   Step 1: Write the system of equations. Grocery store: y = 7x + 35 Wholesale market: y = 9x + 15 Step 2: Set the equations equal to find the intersection. 7x + 35 = 9x + 15 Step 3: Solve for x. 35 - 15 = 9x - 7x 20 = 2x x = 20 / 2 x = 10 Step 4: Substitute x = 10 into either equation to find y. Using y = 7x + 35: y = 7(10) + 35 y = 70 + 35 y = 105 Step 5: Write the solution as an ordered pair. The point of intersection is (10, 105). This means at 10 batches of cookies, both options cost $105.