Systems by Elimination

lessonbunny.com

Answer Key & Explanations

Systems by Elimination · Grade 8 · Worksheet 3

1. Emma is planning a community garden and needs to buy soil and compost. A bag of soil costs $4 and a bag of compost costs $6. She needs a total of 25 bags to fill all the garden beds. Her budget for materials is $130. How many bags of soil and how many bags of compost should Emma buy to use exactly her budget and get the right number of bags? Answer: 10 Solution: Let s = number of soil bags, c = number of compost bags Write the equation for total bags: s + c = 25 Write the equation for total cost: 4s + 6c = 130 Multiply the first equation by 4: 4s + 4c = 100 Subtract this from the cost equation: (4s + 6c) - (4s + 4c) = 130 - 100 Simplify: 2c = 30 Solve…
   Full step-by-step solution

   Step 1: Let s = number of soil bags, c = number of compost bags Step 2: Write the equation for total bags: s + c = 25 Step 3: Write the equation for total cost: 4s + 6c = 130 Step 4: Multiply the first equation by 4: 4s + 4c = 100 Step 5: Subtract this from the cost equation: (4s + 6c) - (4s + 4c) = 130 - 100 Step 6: Simplify: 2c = 30 Step 7: Solve for c: c = 15 Step 8: Substitute back into first equation: s + 15 = 25 Step 9: Solve for s: s = 10 Emma should buy 10 bags of soil and 15 bags of compost.

2. Liam is organizing a school fundraiser and needs to determine how many adult tickets and student tickets were sold. The total number of tickets sold was 120. Adult tickets cost $8 each, student tickets cost $5 each, and the total revenue collected was $780. How many adult tickets and how many student tickets were sold? Answer: 60 adult tickets and 60 student tickets Solution: Let A = number of adult tickets sold Let S = number of student tickets sold Write the equations from the problem.
   Full step-by-step solution

   Let’s define variables: Let A = number of adult tickets sold Let S = number of student tickets sold --- **Step 1: Write the equations from the problem.** From the total number of tickets: A + S = 120 ...(1) From the total revenue: Adult tickets cost $8 → revenue from adults = 8A Student tickets cost $5 → revenue from students = 5S Total revenue = 8A + 5S = 780 ...(2) --- **Step 2: Solve the system of equations.** From equation (1): S = 120 - A Substitute into equation (2): 8A + 5(120 - A) = 780 8A + 600 - 5A = 780 (8A - 5A) + 600 = 780 3A + 600 = 780 --- **Step 3: Solve for A.** 3A = 780 - 600 3A = 180 A = 180 / 3 A = 60 --- **Step 4: Solve for S.** S = 120 - A S = 120 - 60 S = 60 --- **Step 5: Check the solution.** Total tickets: 60 + 60 = 120 ✓ Total revenue: 8×60 + 5×60 = 480 + 300 = 780 ✓ --- **Final answer:** Adult tickets: 60 Student tickets: 60

3. Olivia is organizing a school bake sale and selling two types of items: cupcakes and cookies. A cupcake costs $3 and a cookie costs $1. On the first day, she sold a total of 45 items and collected $87. How many cupcakes and how many cookies did Olivia sell? Write your answer as an ordered pair (cupcakes, cookies). Answer: (21, 24) Solution: Let x = number of cupcakes, y = number of cookies.
   Full step-by-step solution

   Let x = number of cupcakes, y = number of cookies. Equation 1 (total items): x + y = 45 Equation 2 (total money): 3x + 1y = 87 To eliminate y, subtract Equation 1 from Equation 2: (3x + y) - (x + y) = 87 - 45 3x + y - x - y = 42 2x = 42 x = 21 Substitute x = 21 into Equation 1: 21 + y = 45 y = 24 Olivia sold 21 cupcakes and 24 cookies. The answer is (21, 24).

4. Emma is managing a community garden and needs to buy tomato and pepper plants. Tomato plants cost $4 each and pepper plants cost $6 each. She has a budget of $120 and wants to buy exactly 25 plants. How many tomato plants and how many pepper plants should Emma buy to spend exactly her budget and get the right number of plants? Answer: (15, 10) Solution: The elimination method for solving systems of equations involves adding or subtracting equations to eliminate one variable.
   Full step-by-step solution

   The elimination method for solving systems of equations involves adding or subtracting equations to eliminate one variable. This works because if two quantities are equal, adding or subtracting them from other equal quantities maintains the equality. Once one variable is eliminated, you can solve for the remaining variable and then substitute back to find the other.

5. Emma is mixing two types of juice for a school event. She has cranberry juice that costs $4 per liter and orange juice that costs $2 per liter. She wants to make 20 liters of a fruit punch mixture that costs $3 per liter. How many liters of cranberry juice and how many liters of orange juice should Emma use? Answer: 10 Solution: Let c = liters of cranberry juice, o = liters of orange juice Write the volume equation: c + o = 20 Write the cost equation: 4c + 2o = 3 × 20 = 60 Multiply the first equation by 2: 2c + 2o = 40 Subtract this from the cost equation: (4c + 2o) - (2c + 2o) = 60 - 40 Simplify: 2c = 20 Solve for c: c…
   Full step-by-step solution

   Step 1: Let c = liters of cranberry juice, o = liters of orange juice Step 2: Write the volume equation: c + o = 20 Step 3: Write the cost equation: 4c + 2o = 3 × 20 = 60 Step 4: Multiply the first equation by 2: 2c + 2o = 40 Step 5: Subtract this from the cost equation: (4c + 2o) - (2c + 2o) = 60 - 40 Step 6: Simplify: 2c = 20 Step 7: Solve for c: c = 10 Step 8: Substitute back: 10 + o = 20, so o = 10 Emma should use 10 liters of cranberry juice and 10 liters of orange juice.

6. Matiu is helping to organize a school sports day and needs to buy two types of balls: soccer balls and basketballs. Each soccer ball costs $22 and each basketball costs $17. The school needs a total of 28 balls, and the total cost for all the balls is $536. How many soccer balls and how many basketballs does Matiu need to buy? Write your answer as an ordered pair (soccer balls, basketballs). Answer: (12, 16) Solution: Let x = number of soccer balls, y = number of basketballs. x + y = 28 (total number of balls) 22x + 17y = 536 (total cost) Use elimination.
   Full step-by-step solution

   Step 1: Let x = number of soccer balls, y = number of basketballs. Step 2: Write the equations: x + y = 28 (total number of balls) 22x + 17y = 536 (total cost) Step 3: Use elimination. Multiply the first equation by 17 to match the y coefficients: 17x + 17y = 476 22x + 17y = 536 Step 4: Subtract the first new equation from the second: (22x + 17y) - (17x + 17y) = 536 - 476 5x = 60 x = 12 Step 5: Substitute x = 12 into x + y = 28: 12 + y = 28 y = 16 Step 6: The answer is (12, 16), meaning 12 soccer balls and 16 basketballs.

7. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A second triangle is drawn with vertices at (0,0), (9,0), and (9,12). Are these triangles similar? If so, what is the scale factor from the first triangle to the second? Answer: yes, 1.5 Solution: We have two right triangles. First triangle vertices: A(0,0), B(6,0), C(6,8) Second triangle vertices: A'(0,0), B'(9,0), C'(9,12) From A(0,0) to B(6,0): length AB = 6 From B(6,0) to C(6,8): length BC = 8 From A(0,0) to C(6,8): length AC = sqrt((6-0)^2 + (8-0)^2) = sqrt(36 + 64) = sqrt(100) = 10…
   Full step-by-step solution

   Step 1: Understand the problem We have two right triangles. First triangle vertices: A(0,0), B(6,0), C(6,8) Second triangle vertices: A'(0,0), B'(9,0), C'(9,12) Step 2: Determine side lengths of the first triangle From A(0,0) to B(6,0): length AB = 6 From B(6,0) to C(6,8): length BC = 8 From A(0,0) to C(6,8): length AC = sqrt((6-0)^2 + (8-0)^2) = sqrt(36 + 64) = sqrt(100) = 10 So first triangle sides: 6, 8, 10 Step 3: Determine side lengths of the second triangle From A'(0,0) to B'(9,0): length A'B' = 9 From B'(9,0) to C'(9,12): length B'C' = 12 From A'(0,0) to C'(9,12): length A'C' = sqrt((9-0)^2 + (12-0)^2) = sqrt(81 + 144) = sqrt(225) = 15 So second triangle sides: 9, 12, 15 Step 4: Check if triangles are similar Triangles are similar if their corresponding sides are in proportion. Let's compare each side: First triangle sides: 6, 8, 10 Second triangle sides: 9, 12, 15 Check ratios: 9/6 = 1.5 12/8 = 1.5 15/10 = 1.5 All three ratios are equal to 1.5, so the triangles are similar. Step 5: Determine the scale factor Since all side ratios are 1.5, the scale factor from the first triangle to the second triangle is 1.5. Final answer: yes, 1.5