Systems by Elimination

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Answer Key & Explanations

Systems by Elimination · Grade 8 · Worksheet 1

1. Liam is planning a school fundraiser and needs to figure out how many chocolate bars and bags of popcorn to sell. He knows that chocolate bars sell for $2 each and popcorn bags sell for $3 each. His goal is to raise exactly $120. However, he also has a limit on the total number of items he can sell, which is 50. How many chocolate bars and how many bags of popcorn must Liam sell to meet both his revenue goal and his item limit? Answer: 30 chocolate bars and 20 popcorn bags Solution: c = number of chocolate bars p = number of popcorn bags 1. Each chocolate bar sells for $2, each popcorn bag sells for $3, and total revenue is $120. Equation 1: 2c + 3p = 120 2.
   Full step-by-step solution

   Let's define variables first. Let: c = number of chocolate bars p = number of popcorn bags From the problem: 1. Each chocolate bar sells for $2, each popcorn bag sells for $3, and total revenue is $120. Equation 1: 2c + 3p = 120 2. Total items sold is 50. Equation 2: c + p = 50 --- **Step 1: Solve for one variable from Equation 2** From c + p = 50, we get: c = 50 - p --- **Step 2: Substitute into Equation 1** Substitute c = 50 - p into 2c + 3p = 120: 2(50 - p) + 3p = 120 --- **Step 3: Simplify and solve for p** 100 - 2p + 3p = 120 100 + p = 120 p = 120 - 100 p = 20 --- **Step 4: Find c** c = 50 - p = 50 - 20 = 30 --- **Step 5: Verify** Revenue: 2(30) + 3(20) = 60 + 60 = 120 ✔ Total items: 30 + 20 = 50 ✔ --- **Final answer:** Liam must sell 30 chocolate bars and 20 popcorn bags.

2. 3x + 2y = 17; 5x - 2y = 7 Answer: x = 3, y = 4 Solution: (3x + 2y) + (5x - 2y) = 17 + 7 3x + 5x + 2y - 2y = 24 8x = 24 x = 24 ÷ 8 x = 3 Substitute x = 3 into the first equation 3(3) + 2y = 17 9 + 2y = 17 2y = 17 - 9 2y = 8 y = 8 ÷ 2 y = 4 5(3) - 2(4) = 15 - 8 = 7 ✓ The solution is x = 3, y = 4.
   Full step-by-step solution

   Step 1: Add the two equations together to eliminate y (3x + 2y) + (5x - 2y) = 17 + 7 3x + 5x + 2y - 2y = 24 8x = 24 Step 2: Solve for x x = 24 ÷ 8 x = 3 Step 3: Substitute x = 3 into the first equation 3(3) + 2y = 17 9 + 2y = 17 Step 4: Solve for y 2y = 17 - 9 2y = 8 y = 8 ÷ 2 y = 4 Step 5: Verify the solution in the second equation 5(3) - 2(4) = 15 - 8 = 7 ✓ The solution is x = 3, y = 4.

3. 3x + 2y = 19; 5x - 2y = 25 Answer: x = 5.5, y = 1.25 Solution: 3x + 2y = 19 5x - 2y = 25 (3x + 2y) + (5x - 2y) = 19 + 25 3x + 5x + 2y - 2y = 44 8x = 44 x = 44 ÷ 8 x = 5.5 Substitute x = 5.5 into the first equation: 3(5.5) + 2y = 19 16.5 + 2y = 19 2y = 19 - 16.5 2y = 2.5 y = 2.5 ÷ 2 y = 1.25 5(5.5) - 2(1.25) = 27.5 - 2.5 = 25 ✓ The solution is x = 5.5, y = 1.25
   Full step-by-step solution

   Step 1: Write the system: 3x + 2y = 19 5x - 2y = 25 Step 2: Add the equations to eliminate y: (3x + 2y) + (5x - 2y) = 19 + 25 3x + 5x + 2y - 2y = 44 8x = 44 Step 3: Solve for x: x = 44 ÷ 8 x = 5.5 Step 4: Substitute x = 5.5 into the first equation: 3(5.5) + 2y = 19 16.5 + 2y = 19 Step 5: Solve for y: 2y = 19 - 16.5 2y = 2.5 y = 2.5 ÷ 2 y = 1.25 Step 6: Verify with second equation: 5(5.5) - 2(1.25) = 27.5 - 2.5 = 25 ✓ The solution is x = 5.5, y = 1.25

4. Sophia is designing a rectangular garden on a coordinate grid, with vertices at (0, 0), (8, 0), (8, 12), and (0, 12). She wants to add a diagonal pathway from the bottom-left corner to the top-right corner, and another pathway from the bottom-right corner to the top-left corner. The pathways cross at the center of the rectangle. What is the point of intersection where the two diagonals cross? Answer: (4, 6) Solution: Identify the endpoints of one diagonal, say from (0, 0) to (8, 12). The midpoint formula is ((x1 + x2)/2, (y1 + y2)/2). Substitute the coordinates: ((0 + 8)/2, (0 + 12)/2).
   Full step-by-step solution

   Step 1: Identify the endpoints of one diagonal, say from (0, 0) to (8, 12). Step 2: The midpoint formula is ((x1 + x2)/2, (y1 + y2)/2). Step 3: Substitute the coordinates: ((0 + 8)/2, (0 + 12)/2). Step 4: Simplify: (8/2, 12/2) = (4, 6). Step 5: The diagonals of a rectangle intersect at their common midpoint, so the intersection point is (4, 6). The answer is (4, 6).

5. Liam is planning a school fundraiser selling T-shirts and hats. The T-shirts cost $8 each and the hats cost $5 each. On the first day, Liam sold 12 items total and collected $81. How many T-shirts and how many hats did Liam sell? Write your answer as an ordered pair (T-shirts, hats). Answer: (7, 5) Solution: Let T = number of T-shirts sold Let H = number of hats sold Write the equations from the problem.
   Full step-by-step solution

   Let T = number of T-shirts sold Let H = number of hats sold Step 1: Write the equations from the problem. From the total items sold: T + H = 12 From the total money collected (T-shirts cost $8, hats cost $5): 8T + 5H = 81 Step 2: Solve the first equation for one variable. T + H = 12 T = 12 - H Step 3: Substitute T = 12 - H into the second equation. 8(12 - H) + 5H = 81 Step 4: Simplify and solve for H. 96 - 8H + 5H = 81 96 - 3H = 81 Subtract 96 from both sides: -3H = 81 - 96 -3H = -15 Divide both sides by -3: H = 5 Step 5: Find T using T = 12 - H. T = 12 - 5 T = 7 Step 6: Check the solution. T-shirts: 7 × $8 = $56 Hats: 5 × $5 = $25 Total money: $56 + $25 = $81 Total items: 7 + 5 = 12 Both conditions match. Final answer: (7, 5)

6. Hana is organizing a school science fair and needs to buy two types of microscopes: basic and advanced. Basic microscopes cost $40 each and advanced microscopes cost $60 each. She needs to buy a total of 30 microscopes and has a budget of $1,400. How many basic microscopes and how many advanced microscopes should Hana buy to spend exactly her budget? Answer: (20, 10) Solution: Let x = number of basic microscopes, y = number of advanced microscopes.
   Full step-by-step solution

   Let x = number of basic microscopes, y = number of advanced microscopes. Equation 1 (total number): x + y = 30 Equation 2 (total cost): 40x + 60y = 1400 Multiply Equation 1 by 40 to align coefficients for x: 40(x + y) = 40(30) 40x + 40y = 1200 Subtract this new equation from Equation 2: (40x + 60y) - (40x + 40y) = 1400 - 1200 20y = 200 y = 10 Substitute y = 10 into Equation 1: x + 10 = 30 x = 20 Hana should buy 20 basic microscopes and 10 advanced microscopes. Answer: (20, 10)

7. Mere is helping to organize the school's annual sports day and needs to buy two types of medals: gold and silver. Gold medals cost $8 each and silver medals cost $4 each. She has a budget of $200 for medals, and she needs exactly 40 medals in total to award all the winners. How many gold medals and how many silver medals should Mere buy to spend exactly her budget and get the right number of medals? Answer: (10, 30) Solution: Let g = number of gold medals and s = number of silver medals. Equation 1 (total medals): g + s = 40 Equation 2 (total cost): 8g + 4s = 200 Use elimination.
   Full step-by-step solution

   Step 1: Let g = number of gold medals and s = number of silver medals. Step 2: Write the equations: Equation 1 (total medals): g + s = 40 Equation 2 (total cost): 8g + 4s = 200 Step 3: Use elimination. Multiply Equation 1 by 4 to align the s coefficients: 4(g + s) = 4(40) becomes 4g + 4s = 160 Step 4: Subtract this new equation from Equation 2 to eliminate s: (8g + 4s) - (4g + 4s) = 200 - 160 8g + 4s - 4g - 4s = 40 4g = 40 Step 5: Solve for g: g = 40 / 4 = 10 gold medals. Step 6: Substitute g = 10 into Equation 1: 10 + s = 40, so s = 30 silver medals. Step 7: Check in Equation 2: 8(10) + 4(30) = 80 + 120 = 200. Correct. The answer is (10 gold medals, 30 silver medals).