Systems by Graphing

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Systems by Graphing · Grade 8 · Worksheet 3

1. Maya is organizing a community garden and needs to plan the planting of tomatoes and peppers. She has space for exactly 30 plants total. Tomatoes require 2 square feet each and peppers require 1 square foot each, and the total garden space is 45 square feet. If x represents the number of tomato plants and y represents the number of pepper plants, how many of each should Maya plant to use all her space efficiently? Answer: (15, 15) Solution: When solving systems of equations by graphing, you represent each condition as a line on a coordinate plane. The point where the lines intersect gives the values that satisfy both conditions simultaneously.
   Full step-by-step solution

   When solving systems of equations by graphing, you represent each condition as a line on a coordinate plane. The point where the lines intersect gives the values that satisfy both conditions simultaneously. This method is useful for comparing two constraints, like quantity and resource usage.

2. y = 2x - 3 and y = -x + 6 Answer: (3, 3) Solution: Set the equations equal to find the x-coordinate: 2x - 3 = -x + 6 Add x to both sides: 3x - 3 = 6 Add 3 to both sides: 3x = 9 Divide by 3: x = 3 Substitute x = 3 into either equation to find y: y = 2(3) - 3 = 6 - 3 = 3 The solution is the ordered pair (3, 3) The answer is (3, 3).
   Full step-by-step solution

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

3. A rectangular garden is drawn on a coordinate plane with corners at (2, 1), (8, 1), (8, 5), and (2, 5). A diagonal path is drawn from (2, 1) to (8, 5), creating two triangular sections. What is the area of one of these triangular sections? Answer: 12 Solution: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) The diagonal is drawn from A(2, 1) to C(8, 5). This diagonal splits the rectangle into two congruent triangles.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangle with vertices: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) The diagonal is drawn from A(2, 1) to C(8, 5). This diagonal splits the rectangle into two congruent triangles. --- **Step 2: Find the area of the rectangle** Length along x-axis: from x = 2 to x = 8 → length = 8 - 2 = 6 Height along y-axis: from y = 1 to y = 5 → height = 5 - 1 = 4 Area of rectangle = length × height = 6 × 4 = 24 --- **Step 3: Area of one triangle** The diagonal divides the rectangle into two triangles of equal area. Area of one triangle = (Area of rectangle) / 2 = 24 / 2 = 12 --- **Step 4: Conclusion** The area of one triangular section is 12. --- **Final answer:** 12

4. Liam is planning a school fundraiser and needs to decide between two catering options for the event. Option A charges a $150 setup fee plus $8 per person. Option B charges a $50 setup fee plus $12 per person. Liam wants to know for how many guests the total cost would be the same for both options. Write your answer as a whole number. Answer: 25 Solution: Let’s define the total cost for each option. Setup fee = $150 Cost per person = $8 Total cost for x guests = 150 + 8x Setup fee = $50 Cost per person = $12 Total cost for x guests = 50 + 12x We want the number of guests x where both costs are equal: 150 + 8x = 50 + 12x Subtract 50 from both…
   Full step-by-step solution

   Let’s define the total cost for each option. Option A: Setup fee = $150 Cost per person = $8 Total cost for x guests = 150 + 8x Option B: Setup fee = $50 Cost per person = $12 Total cost for x guests = 50 + 12x We want the number of guests x where both costs are equal: 150 + 8x = 50 + 12x Step 1: Subtract 50 from both sides 150 − 50 + 8x = 50 − 50 + 12x 100 + 8x = 12x Step 2: Subtract 8x from both sides 100 + 8x − 8x = 12x − 8x 100 = 4x Step 3: Divide both sides by 4 100 / 4 = x x = 25 So, for 25 guests, the total cost is the same for both options.

5. 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.

6. Liam is planning a school fundraiser and needs to decide between selling chocolate bars and cookies. He calculates that selling chocolate bars gives him a profit of $2 per bar, while cookies give him a profit of $1.50 per cookie. He wants to graph these two profit scenarios to see where they are equal. The equation for chocolate bar profit is y = 2x, and for cookie profit is y = 1.5x + 10, where x represents the number of items sold and y represents the total profit in dollars. After graphing both lines, at what point do the two profit lines intersect? Answer: (20, 40) Solution: 1. Chocolate bar profit: y = 2x 2. Cookie profit: y = 1.5x + 10 We want to find the point (x, y) where the two lines intersect.
   Full step-by-step solution

   We are given two equations: 1. Chocolate bar profit: y = 2x 2. Cookie profit: y = 1.5x + 10 We want to find the point (x, y) where the two lines intersect. At the intersection point, the y-values are equal, so we set the right-hand sides equal to each other. Step 1: Set the equations equal. 2x = 1.5x + 10 Step 2: Subtract 1.5x from both sides to get the x terms on one side. 2x - 1.5x = 10 0.5x = 10 Step 3: Solve for x. x = 10 / 0.5 x = 20 Step 4: Substitute x = 20 into one of the original equations to find y. Using y = 2x: y = 2 * 20 y = 40 Step 5: Interpret the result. The intersection point is (20, 40). This means when 20 items are sold, the total profit for both chocolate bars and cookies is $40. Final answer: (20, 40)