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Systems by Substitution

Grade 8 · Algebra · Worksheet 2

  1. Mason is organizing a school carnival and needs to determine how many game booths and food stalls to set up. Each game booth requires 2 volunteers to run it, and each food stall requires 4 volunteers. Mason has a total of 48 volunteers available. Additionally, he wants to have exactly 18 booths and stalls combined. How many game booths and how many food stalls should Mason set up? Answer: ______________
  2. y = 4x - 9; 5x + 2y = 21; x = ? Answer: ______________
  3. Mere is helping to organize a school craft fair. She needs to buy two types of yarn: wool yarn costs $6 per skein, and cotton yarn costs $4 per skein. She has a budget of $108 and wants to buy exactly 22 skeins of yarn in total. How many skeins of wool yarn and how many skeins of cotton yarn should Mere buy? Answer: ______________
  4. 2x + 3y = 23; y = 4x - 11; x = ? Answer: ______________
  5. 3x - 2y = 16; y = 2x - 9; x = ? Answer: ______________
  6. Liam is planning a school fundraiser and needs to determine how many cupcakes and cookies to bake. He knows that cupcakes require 2 cups of flour each and cookies require 1.5 cups of flour each. He has 30 cups of flour available. Additionally, the total number of baked goods (cupcakes + cookies) should be 20. Let c represent the number of cupcakes and k represent the number of cookies. Write and solve a system of equations to find how many cupcakes and cookies Liam should bake. Answer: ______________
  7. 2x + y = 7; y = 3x - 1 Answer: ______________
  8. 2x + 3y = 12, y = x - 1 Answer: ______________
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Answer Key & Explanations

Systems by Substitution · Grade 8 · Worksheet 2

  1. Mason is organizing a school carnival and needs to determine how many game booths and food stalls to set up. Each game booth requires 2 volunteers to run it, and each food stall requires 4 volunteers. Mason has a total of 48 volunteers available. Additionally, he wants to have exactly 18 booths and stalls combined. How many game booths and how many food stalls should Mason set up? Answer: 12 game booths and 6 food stalls Solution: Let g = number of game booths and f = number of food stalls.
    Full step-by-step solution

    Let g = number of game booths and f = number of food stalls. Equation 1 (total booths/stalls): g + f = 18 Equation 2 (total volunteers): 2g + 4f = 48 Solve Equation 1 for g: g = 18 - f Substitute into Equation 2: 2(18 - f) + 4f = 48 36 - 2f + 4f = 48 36 + 2f = 48 2f = 12 f = 6 Now substitute f = 6 back into g = 18 - f: g = 18 - 6 g = 12 So Mason should set up 12 game booths and 6 food stalls. Check: 12 + 6 = 18 booths/stalls total. 2(12) + 4(6) = 24 + 24 = 48 volunteers. Both conditions are satisfied. The answer is 12 game booths and 6 food stalls.

  2. y = 4x - 9; 5x + 2y = 21; x = ? Answer: 3 Solution: Start with the system: y = 4x - 9 and 5x + 2y = 21. Substitute y = 4x - 9 into the second equation: 5x + 2(4x - 9) = 21. Distribute the 2: 5x + 8x - 18 = 21.
    Full step-by-step solution

    Step 1: Start with the system: y = 4x - 9 and 5x + 2y = 21. Step 2: Substitute y = 4x - 9 into the second equation: 5x + 2(4x - 9) = 21. Step 3: Distribute the 2: 5x + 8x - 18 = 21. Step 4: Combine like terms: 13x - 18 = 21. Step 5: Add 18 to both sides: 13x = 39. Step 6: Divide both sides by 13: x = 3. The value of x is 3.

  3. Mere is helping to organize a school craft fair. She needs to buy two types of yarn: wool yarn costs $6 per skein, and cotton yarn costs $4 per skein. She has a budget of $108 and wants to buy exactly 22 skeins of yarn in total. How many skeins of wool yarn and how many skeins of cotton yarn should Mere buy? Answer: wool: 10, cotton: 12 Solution: Let w = number of wool skeins, c = number of cotton skeins.
    Full step-by-step solution

    Let w = number of wool skeins, c = number of cotton skeins. Equation 1 (total skeins): w + c = 22 Equation 2 (total cost): 6w + 4c = 108 Solve equation 1 for c: c = 22 - w Substitute into equation 2: 6w + 4(22 - w) = 108 Expand: 6w + 88 - 4w = 108 Combine like terms: 2w + 88 = 108 Subtract 88 from both sides: 2w = 20 Divide by 2: w = 10 Substitute back: c = 22 - 10 = 12 Mere should buy 10 skeins of wool yarn and 12 skeins of cotton yarn.

  4. 2x + 3y = 23; y = 4x - 11; x = ? Answer: 4 Solution: Start with the system: 2x + 3y = 23 and y = 4x - 11. Substitute y = 4x - 11 into the first equation: 2x + 3(4x - 11) = 23. Distribute the 3: 2x + 12x - 33 = 23.
    Full step-by-step solution

    Step 1: Start with the system: 2x + 3y = 23 and y = 4x - 11. Step 2: Substitute y = 4x - 11 into the first equation: 2x + 3(4x - 11) = 23. Step 3: Distribute the 3: 2x + 12x - 33 = 23. Step 4: Combine like terms: 14x - 33 = 23. Step 5: Add 33 to both sides: 14x = 56. Step 6: Divide both sides by 14: x = 4. The value of x is 4.

  5. 3x - 2y = 16; y = 2x - 9; x = ? Answer: 2 Solution: Start with the system: 3x - 2y = 16 and y = 2x - 9. Substitute (2x - 9) for y in the first equation: 3x - 2(2x - 9) = 16. Distribute the -2: 3x - 4x + 18 = 16.
    Full step-by-step solution

    Step 1: Start with the system: 3x - 2y = 16 and y = 2x - 9. Step 2: Substitute (2x - 9) for y in the first equation: 3x - 2(2x - 9) = 16. Step 3: Distribute the -2: 3x - 4x + 18 = 16. Step 4: Combine like terms: -x + 18 = 16. Step 5: Subtract 18 from both sides: -x = -2. Step 6: Multiply both sides by -1: x = 2. The value of x is 2.

  6. Liam is planning a school fundraiser and needs to determine how many cupcakes and cookies to bake. He knows that cupcakes require 2 cups of flour each and cookies require 1.5 cups of flour each. He has 30 cups of flour available. Additionally, the total number of baked goods (cupcakes + cookies) should be 20. Let c represent the number of cupcakes and k represent the number of cookies. Write and solve a system of equations to find how many cupcakes and cookies Liam should bake. Answer: c = 12, k = 8 Solution: The substitution method for solving systems of equations involves isolating one variable in one equation and then substituting that expression into the other equation.
    Full step-by-step solution

    The substitution method for solving systems of equations involves isolating one variable in one equation and then substituting that expression into the other equation. This transforms the system into a single equation with one variable. After solving for that variable, you substitute the value back into one of the original equations to find the other variable. This method is particularly useful when one equation is already solved for a variable or can be easily manipulated to do so.

  7. 2x + y = 7; y = 3x - 1 Answer: x = 1.6, y = 3.8 Solution: 1) 2x + y = 7 2) y = 3x - 1 Substitute equation (2) into equation (1). Since equation (2) says y = 3x - 1, we can replace y in equation (1) with 3x - 1. So equation (1) becomes: 2x + (3x - 1) = 7 Combine like terms.
    Full step-by-step solution

    We are given the system of equations: 1) 2x + y = 7 2) y = 3x - 1 Step 1: Substitute equation (2) into equation (1). Since equation (2) says y = 3x - 1, we can replace y in equation (1) with 3x - 1. So equation (1) becomes: 2x + (3x - 1) = 7 Step 2: Combine like terms. 2x + 3x - 1 = 7 5x - 1 = 7 Step 3: Add 1 to both sides. 5x - 1 + 1 = 7 + 1 5x = 8 Step 4: Divide both sides by 5. x = 8/5 x = 1.6 Step 5: Substitute x = 1.6 into equation (2) to find y. y = 3(1.6) - 1 y = 4.8 - 1 y = 3.8 Step 6: Check in equation (1). 2(1.6) + 3.8 = 3.2 + 3.8 = 7 ✓ Final answer: x = 1.6, y = 3.8

  8. 2x + 3y = 12, y = x - 1 Answer: x = 3, y = 2 Solution: 1) 2x + 3y = 12 2) y = x - 1 Substitute equation (2) into equation (1). Since y = x - 1, we replace y in equation (1) with (x - 1): 2x + 3(x - 1) = 12 Simplify and solve for x.
    Full step-by-step solution

    We are given the system of equations: 1) 2x + 3y = 12 2) y = x - 1 Step 1: Substitute equation (2) into equation (1). Since y = x - 1, we replace y in equation (1) with (x - 1): 2x + 3(x - 1) = 12 Step 2: Simplify and solve for x. First, distribute the 3: 2x + 3x - 3 = 12 Combine like terms: 5x - 3 = 12 Add 3 to both sides: 5x = 15 Divide both sides by 5: x = 3 Step 3: Substitute x = 3 into equation (2) to find y. y = 3 - 1 y = 2 Step 4: Check the solution in equation (1). 2(3) + 3(2) = 6 + 6 = 12, which matches. Final answer: x = 3, y = 2