Systems by Elimination

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Answer Key & Explanations

Systems by Elimination · Grade 8 · Worksheet 1

1. 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).

2. 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.

3. A triangular garden is drawn on a coordinate plane with vertices at (0,0), (6,0), and (3,4). A second triangle is drawn with vertices at (0,0), (9,0), and (4.5,6). Are these triangles similar? If so, what is the scale factor from the first triangle to the second? Answer: 1.5 Solution: - Side A: between (0,0) and (6,0) = 6 units - Side B: between (0,0) and (3,4) = sqrt((3-0)^2 + (4-0)^2) = sqrt(9 + 16) = sqrt(25) = 5 units - Side C: between (6,0) and (3,4) = sqrt((3-6)^2 + (4-0)^2) = sqrt(9 + 16) = sqrt(25) = 5 units - Side A': between (0,0) and (9,0) = 9 units - Side B':…
   Full step-by-step solution

   Step 1: Calculate side lengths of first triangle - Side A: between (0,0) and (6,0) = 6 units - Side B: between (0,0) and (3,4) = sqrt((3-0)^2 + (4-0)^2) = sqrt(9 + 16) = sqrt(25) = 5 units - Side C: between (6,0) and (3,4) = sqrt((3-6)^2 + (4-0)^2) = sqrt(9 + 16) = sqrt(25) = 5 units Step 2: Calculate side lengths of second triangle - Side A': between (0,0) and (9,0) = 9 units - Side B': between (0,0) and (4.5,6) = sqrt((4.5-0)^2 + (6-0)^2) = sqrt(20.25 + 36) = sqrt(56.25) = 7.5 units - Side C': between (9,0) and (4.5,6) = sqrt((4.5-9)^2 + (6-0)^2) = sqrt(20.25 + 36) = sqrt(56.25) = 7.5 units Step 3: Compare corresponding side ratios - A'/A = 9/6 = 1.5 - B'/B = 7.5/5 = 1.5 - C'/C = 7.5/5 = 1.5 Step 4: Since all corresponding side ratios are equal (1.5), the triangles are similar Step 5: The scale factor from the first triangle to the second is 1.5 The answer is 1.5.

4. Charlotte is organizing a school recycling competition. She has two types of collection bins: small bins that hold 7 pounds of paper each and large bins that hold 12 pounds of paper each. On Tuesday, Charlotte collected a total of 97 pounds of paper using 11 bins. How many small bins and how many large bins did Charlotte use? Write your answer as an ordered pair (small bins, large bins). Answer: (7, 4) Solution: Let s = number of small bins and l = number of large bins.
   Full step-by-step solution

   Let s = number of small bins and l = number of large bins. Equation 1 (total bins): s + l = 11 Equation 2 (total weight): 7s + 12l = 97 To eliminate s, multiply Equation 1 by 7: 7(s + l) = 7(11) 7s + 7l = 77 Now subtract this from Equation 2: (7s + 12l) - (7s + 7l) = 97 - 77 5l = 20 l = 4 Substitute l = 4 into Equation 1: s + 4 = 11 s = 7 Charlotte used 7 small bins and 4 large bins. The answer is (7, 4).

5. 3x + 2y = 18; 5x - 2y = 14 Answer: x = 4, y = 3 Solution: (3x + 2y) + (5x - 2y) = 18 + 14 3x + 5x + 2y - 2y = 32 8x = 32 8x = 32 x = 32 ÷ 8 x = 4 Substitute x = 4 into the first equation 3(4) + 2y = 18 12 + 2y = 18 2y = 18 - 12 2y = 6 y = 6 ÷ 2 y = 3 5(4) - 2(3) = 20 - 6 = 14 ✓ The solution is x = 4, y = 3.
   Full step-by-step solution

   Step 1: Add the two equations together to eliminate y (3x + 2y) + (5x - 2y) = 18 + 14 3x + 5x + 2y - 2y = 32 8x = 32 Step 2: Solve for x 8x = 32 x = 32 ÷ 8 x = 4 Step 3: Substitute x = 4 into the first equation 3(4) + 2y = 18 12 + 2y = 18 Step 4: Solve for y 2y = 18 - 12 2y = 6 y = 6 ÷ 2 y = 3 Step 5: Verify with second equation 5(4) - 2(3) = 20 - 6 = 14 ✓ The solution is x = 4, y = 3.

6. Mason is helping to organize a school science fair. He needs to buy two types of materials: packs of poster boards and packs of markers. A pack of poster boards costs $9, and a pack of markers costs $7. Mason buys a total of 18 packs of materials and spends exactly $142. How many packs of poster boards and how many packs of markers did Mason buy? Write your answer as an ordered pair (poster boards, markers). Answer: (13, 5) Solution: Let p = number of packs of poster boards, m = number of packs of markers. Write two equations from the problem. Total packs: p + m = 18 Total cost: 9p + 7m = 142 To use elimination, we can eliminate m.
   Full step-by-step solution

   Step 1: Let p = number of packs of poster boards, m = number of packs of markers. Step 2: Write two equations from the problem. Total packs: p + m = 18 Total cost: 9p + 7m = 142 Step 3: To use elimination, we can eliminate m. Multiply the first equation by 7 to make the coefficients of m the same. 7(p + m) = 7(18) => 7p + 7m = 126 Step 4: Subtract this new equation from the cost equation to eliminate m. (9p + 7m) - (7p + 7m) = 142 - 126 9p + 7m - 7p - 7m = 16 2p = 16 Step 5: Solve for p. 2p = 16 p = 8 Step 6: Substitute p = 8 into the first equation to find m. p + m = 18 8 + m = 18 m = 10 Step 7: Check the solution in the cost equation. 9(8) + 7(10) = 72 + 70 = 142. Correct. The answer is (8, 10).

7. 2x + 3y = 16; 2x - y = 0 Answer: x = 2, y = 4 Solution: (1) 2x + 3y = 16 (2) 2x - y = 0 Solve equation (2) for y. From equation (2): 2x - y = 0 Add y to both sides: 2x = y So y = 2x. Substitute y = 2x into equation (1).
   Full step-by-step solution

   We are given the system of equations: (1) 2x + 3y = 16 (2) 2x - y = 0 Step 1: Solve equation (2) for y. From equation (2): 2x - y = 0 Add y to both sides: 2x = y So y = 2x. Step 2: Substitute y = 2x into equation (1). Equation (1): 2x + 3y = 16 Replace y with 2x: 2x + 3*(2x) = 16 Multiply: 2x + 6x = 16 Combine like terms: 8x = 16 Step 3: Solve for x. Divide both sides by 8: x = 16/8 x = 2. Step 4: Solve for y using y = 2x. y = 2*(2) = 4. Step 5: Check the solution in both original equations. Equation (1): 2*(2) + 3*(4) = 4 + 12 = 16 ✓ Equation (2): 2*(2) - 4 = 4 - 4 = 0 ✓ Final answer: x = 2, y = 4