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Systems by Substitution

Grade 8 · Algebra · Worksheet 3

  1. 3x - 2y = 7; y = 2x - 5; x = ? Answer: ______________
  2. Isabella is organizing the school's recycling drive. She collects aluminum cans and plastic bottles. Each aluminum can weighs 2 ounces and each plastic bottle weighs 7 ounces. She has a total of 42 items collected, and their combined weight is 144 ounces. How many aluminum cans and how many plastic bottles did Isabella collect? Answer: ______________
  3. y = 5x - 10; 3x + 2y = 45; x = ? Answer: ______________
  4. A rectangular pool is drawn on a coordinate grid. The four corners are at A(10, 12), B(22, 12), C(22, 20), and D(10, 20). A straight water line runs from the midpoint of side AB to the midpoint of side CD. The equation of this water line is y = 16. A diagonal cleaning robot path runs from corner A to corner C, with equation y = (2/3)x + (16/3). Using the substitution method, at what coordinate point does the robot path intersect the water line? Answer: ______________
  5. Emma is helping her cousin organize a school bake sale. They are selling two types of cookies: chocolate chip cookies and oatmeal raisin cookies. Each chocolate chip cookie requires 3 ounces of dough, and each oatmeal raisin cookie requires 5 ounces of dough. They have a total of 189 ounces of dough prepared. They also know that they need to make exactly 47 cookies in total to fill all the display trays. How many chocolate chip cookies and how many oatmeal raisin cookies should they make? Answer: ______________
  6. 3x - 2y = 16; y = 2x - 10; x = ? Answer: ______________
  7. 2x + 3y = 12 and y = x - 1, solve for x and y Answer: ______________
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Answer Key & Explanations

Systems by Substitution · Grade 8 · Worksheet 3

  1. 3x - 2y = 7; y = 2x - 5; x = ? Answer: 3 Solution: Start with the system: 3x - 2y = 7 and y = 2x - 5. Substitute (2x - 5) for y in the first equation: 3x - 2(2x - 5) = 7. Distribute the -2: 3x - 4x + 10 = 7.
    Full step-by-step solution

    Step 1: Start with the system: 3x - 2y = 7 and y = 2x - 5. Step 2: Substitute (2x - 5) for y in the first equation: 3x - 2(2x - 5) = 7. Step 3: Distribute the -2: 3x - 4x + 10 = 7. Step 4: Combine like terms: -x + 10 = 7. Step 5: Subtract 10 from both sides: -x = -3. Step 6: Multiply both sides by -1: x = 3. The value of x is 3.

  2. Isabella is organizing the school's recycling drive. She collects aluminum cans and plastic bottles. Each aluminum can weighs 2 ounces and each plastic bottle weighs 7 ounces. She has a total of 42 items collected, and their combined weight is 144 ounces. How many aluminum cans and how many plastic bottles did Isabella collect? Answer: 30 aluminum cans and 12 plastic bottles Solution: Let a = number of aluminum cans, p = number of plastic bottles.
    Full step-by-step solution

    Let a = number of aluminum cans, p = number of plastic bottles. Equation 1 (total items): a + p = 42 Equation 2 (total weight): 2a + 7p = 144 Solve Equation 1 for a: a = 42 - p Substitute into Equation 2: 2(42 - p) + 7p = 144 Simplify: 84 - 2p + 7p = 144 Combine like terms: 84 + 5p = 144 Subtract 84 from both sides: 5p = 60 Divide by 5: p = 12 Substitute p = 12 into a = 42 - p: a = 42 - 12 = 30 Check: 30 cans at 2 oz each = 60 oz; 12 bottles at 7 oz each = 84 oz; total weight = 60 + 84 = 144 oz. Total items = 30 + 12 = 42. Isabella collected 30 aluminum cans and 12 plastic bottles.

  3. y = 5x - 10; 3x + 2y = 45; x = ? Answer: 5 Solution: Start with the system: y = 5x - 10 and 3x + 2y = 45. Substitute y = 5x - 10 into the second equation: 3x + 2(5x - 10) = 45. Distribute the 2: 3x + 10x - 20 = 45.
    Full step-by-step solution

    Step 1: Start with the system: y = 5x - 10 and 3x + 2y = 45. Step 2: Substitute y = 5x - 10 into the second equation: 3x + 2(5x - 10) = 45. Step 3: Distribute the 2: 3x + 10x - 20 = 45. Step 4: Combine like terms: 13x - 20 = 45. Step 5: Add 20 to both sides: 13x = 65. Step 6: Divide both sides by 13: x = 5. The value of x is 5.

  4. A rectangular pool is drawn on a coordinate grid. The four corners are at A(10, 12), B(22, 12), C(22, 20), and D(10, 20). A straight water line runs from the midpoint of side AB to the midpoint of side CD. The equation of this water line is y = 16. A diagonal cleaning robot path runs from corner A to corner C, with equation y = (2/3)x + (16/3). Using the substitution method, at what coordinate point does the robot path intersect the water line? Answer: (16, 16) Solution: Identify the two equations. Water line: y = 16. Robot path: y = (2/3)x + (16/3).
    Full step-by-step solution

    Step 1: Identify the two equations. Water line: y = 16. Robot path: y = (2/3)x + (16/3). Step 2: Substitute y = 16 into the robot path equation: 16 = (2/3)x + (16/3). Step 3: Subtract (16/3) from both sides: 16 - (16/3) = (2/3)x. Write 16 as 48/3: (48/3 - 16/3) = (2/3)x, so (32/3) = (2/3)x. Step 4: Multiply both sides by 3: 32 = 2x. Step 5: Divide both sides by 2: x = 16. Step 6: The intersection point is (16, 16). The answer is (16, 16).

  5. Emma is helping her cousin organize a school bake sale. They are selling two types of cookies: chocolate chip cookies and oatmeal raisin cookies. Each chocolate chip cookie requires 3 ounces of dough, and each oatmeal raisin cookie requires 5 ounces of dough. They have a total of 189 ounces of dough prepared. They also know that they need to make exactly 47 cookies in total to fill all the display trays. How many chocolate chip cookies and how many oatmeal raisin cookies should they make? Answer: 23 chocolate chip cookies and 24 oatmeal raisin cookies Solution: Let c = number of chocolate chip cookies, and o = number of oatmeal raisin cookies.
    Full step-by-step solution

    Let c = number of chocolate chip cookies, and o = number of oatmeal raisin cookies. Equation 1 (total cookies): c + o = 47 Equation 2 (total dough): 3c + 5o = 189 Solve Equation 1 for c: c = 47 - o Substitute into Equation 2: 3(47 - o) + 5o = 189 Simplify: 141 - 3o + 5o = 189 141 + 2o = 189 2o = 48 o = 24 Substitute o = 24 into c = 47 - o: c = 47 - 24 = 23 Check: 23 chocolate chip cookies use 23 * 3 = 69 ounces of dough. 24 oatmeal raisin cookies use 24 * 5 = 120 ounces of dough. Total dough = 69 + 120 = 189 ounces. Total cookies = 23 + 24 = 47. They should make 23 chocolate chip cookies and 24 oatmeal raisin cookies.

  6. 3x - 2y = 16; y = 2x - 10; x = ? Answer: 4 Solution: Start with the system: 3x - 2y = 16 and y = 2x - 10 Substitute y = 2x - 10 into the first equation: 3x - 2(2x - 10) = 16 Distribute the -2: 3x - 4x + 20 = 16 Combine like terms: -x + 20 = 16 Subtract 20 from both sides: -x = -4 Multiply both sides by -1: x = 4 The answer is 4.
    Full step-by-step solution

    Step 1: Start with the system: 3x - 2y = 16 and y = 2x - 10 Step 2: Substitute y = 2x - 10 into the first equation: 3x - 2(2x - 10) = 16 Step 3: Distribute the -2: 3x - 4x + 20 = 16 Step 4: Combine like terms: -x + 20 = 16 Step 5: Subtract 20 from both sides: -x = -4 Step 6: Multiply both sides by -1: x = 4 The answer is 4.

  7. 2x + 3y = 12 and y = x - 1, solve for x and y Answer: x = 3, y = 2 Solution: 1) 2x + 3y = 12 2) y = x - 1 We need to solve for x and y. Substitute y from the second equation into the first equation From equation (2), we have y = x - 1.
    Full step-by-step solution

    We are given two equations: 1) 2x + 3y = 12 2) y = x - 1 We need to solve for x and y. --- **Step 1: Substitute y from the second equation into the first equation** From equation (2), we have y = x - 1. Replace y in equation (1) with (x - 1): 2x + 3(x - 1) = 12 --- **Step 2: Simplify and solve for x** First, distribute 3 into (x - 1): 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: Solve for y** Use y = x - 1: y = 3 - 1 y = 2 --- **Step 4: Check the solution** Plug x = 3, y = 2 into equation (1): 2(3) + 3(2) = 6 + 6 = 12 ✓ Equation (2): 2 = 3 - 1 ✓ --- **Final answer:** x = 3, y = 2