Systems by Substitution

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Systems by Substitution · Grade 8 · Worksheet 1

1. 2x + 3y = 12 and y = x - 1, find x and y Answer: x = 3, y = 2 Solution: 1) 2x + 3y = 12 2) y = x - 1 Substitute the expression for y from equation 2 into equation 1. Since y = x - 1, 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 the expression for y from equation 2 into equation 1. Since 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 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 = x - 1 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

2. y = 3x - 5; 5x + 2y = 45; x = ? Answer: 5 Solution: Start with the system: y = 3x - 5 and 5x + 2y = 45. Substitute y = 3x - 5 into the second equation: 5x + 2(3x - 5) = 45. Distribute the 2: 5x + 6x - 10 = 45.
   Full step-by-step solution

   Step 1: Start with the system: y = 3x - 5 and 5x + 2y = 45. Step 2: Substitute y = 3x - 5 into the second equation: 5x + 2(3x - 5) = 45. Step 3: Distribute the 2: 5x + 6x - 10 = 45. Step 4: Combine like terms: 11x - 10 = 45. Step 5: Add 10 to both sides: 11x = 55. Step 6: Divide both sides by 11: x = 5. The answer is 5.

3. Sophia is helping to organize a school book fair. She needs to order paperback books and hardcover books. Each paperback costs $8 and each hardcover costs $14. She has a total budget of $292 and wants to order exactly 26 books in total. How many paperback books and how many hardcover books should Sophia order? Answer: x = 12 paperback books, y = 14 hardcover books Solution: Let x = number of paperback books and y = number of hardcover books.
   Full step-by-step solution

   Step 1: Let x = number of paperback books and y = number of hardcover books. Step 2: Write the equations based on the problem: Equation 1 (total books): x + y = 26 Equation 2 (total cost): 8x + 14y = 292 Step 3: Solve Equation 1 for x: x = 26 - y Step 4: Substitute x = 26 - y into Equation 2: 8(26 - y) + 14y = 292 Step 5: Distribute: 208 - 8y + 14y = 292 Step 6: Combine like terms: 208 + 6y = 292 Step 7: Subtract 208 from both sides: 6y = 84 Step 8: Divide both sides by 6: y = 14 Step 9: Substitute y = 14 back into x = 26 - y: x = 26 - 14 = 12 Step 10: Check: 12 + 14 = 26 books, and 8(12) + 14(14) = 96 + 196 = 292. The answer is 12 paperback books and 14 hardcover books.

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (0,8). A line representing the hypotenuse has the equation y = (-4/3)x + 8. A second line representing a support beam is drawn from the midpoint of the hypotenuse to the right angle vertex at (0,0). If this support beam has the equation y = (3/4)x, at what coordinate point do these two lines intersect? Answer: (3.84, 2.88) Solution: Set the two equations equal to each other since both equal y at the intersection point: (-4/3)x + 8 = (3/4)x Get all x terms on one side by adding (4/3)x to both sides: 8 = (3/4)x + (4/3)x Find a common denominator to combine the x terms.
   Full step-by-step solution

   Step 1: Set the two equations equal to each other since both equal y at the intersection point: (-4/3)x + 8 = (3/4)x Step 2: Get all x terms on one side by adding (4/3)x to both sides: 8 = (3/4)x + (4/3)x Step 3: Find a common denominator to combine the x terms. The common denominator of 4 and 3 is 12: 8 = (9/12)x + (16/12)x Step 4: Combine the x terms: 8 = (25/12)x Step 5: Multiply both sides by 12/25 to solve for x: x = 8 × (12/25) = 96/25 = 3.84 Step 6: Substitute x = 3.84 back into either original equation to find y. Using y = (3/4)x: y = (3/4) × (96/25) = (288)/(100) = 2.88 Step 7: The intersection point is (3.84, 2.88) The answer is (3.84, 2.88).

5. Liam is planning a school fundraiser and needs to determine how many cupcakes and cookies to sell. He knows that cupcakes sell for $3 each and cookies for $2 each. His goal is to raise exactly $120. He also knows that the total number of baked goods sold will be 50 items. How many cupcakes and how many cookies does Liam need to sell? Answer: 20 cupcakes and 30 cookies Solution: Let c = number of cupcakes Let k = number of cookies Write the equations from the problem. c + k = 50 ... (1) Cupcakes cost $3 each, cookies cost $2 each, total $120: 3c + 2k = 120 ...
   Full step-by-step solution

   Let's define variables first. Let c = number of cupcakes Let k = number of cookies --- **Step 1: Write the equations from the problem.** From the total number of items: c + k = 50 ... (1) From the total money raised: Cupcakes cost $3 each, cookies cost $2 each, total $120: 3c + 2k = 120 ... (2) --- **Step 2: Solve for one variable in terms of the other using equation (1).** From (1): k = 50 - c --- **Step 3: Substitute into equation (2).** 3c + 2(50 - c) = 120 3c + 100 - 2c = 120 (3c - 2c) + 100 = 120 c + 100 = 120 --- **Step 4: Solve for c.** c = 120 - 100 c = 20 --- **Step 5: Solve for k.** k = 50 - c k = 50 - 20 k = 30 --- **Step 6: Check the solution.** Total items: 20 + 30 = 50 ✔ Total money: 3*20 + 2*30 = 60 + 60 = 120 ✔ --- **Final answer:** Liam needs to sell 20 cupcakes and 30 cookies.

6. A science class is studying bacteria growth. The number of bacteria in Culture A can be modeled by the equation y = 3x + 15, where x is time in hours and y is the number of bacteria in thousands. Culture B follows the equation y = 2x + 25. After how many hours will both cultures have the same number of bacteria? Answer: 10 Solution: We are given two equations for the number of bacteria (in thousands) in two cultures: Culture A: y = 3x + 15 Culture B: y = 2x + 25 We want the time x (in hours) when both cultures have the same number of bacteria.
   Full step-by-step solution

   We are given two equations for the number of bacteria (in thousands) in two cultures: Culture A: y = 3x + 15 Culture B: y = 2x + 25 We want the time x (in hours) when both cultures have the same number of bacteria. Step 1: Set the two equations equal to each other because at that moment, y is the same for both. 3x + 15 = 2x + 25 Step 2: Subtract 2x from both sides to get the x terms on one side. 3x - 2x + 15 = 25 x + 15 = 25 Step 3: Subtract 15 from both sides to solve for x. x = 25 - 15 x = 10 Step 4: Interpret the result. After 10 hours, both cultures will have the same number of bacteria. We can check by substituting x = 10 into both equations: For Culture A: y = 3(10) + 15 = 30 + 15 = 45 For Culture B: y = 2(10) + 25 = 20 + 25 = 45 Both give 45 (thousand bacteria), so the answer is correct. Answer: 10 hours

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

8. y = 3x - 7; 5x + 2y = 19; x = ? Answer: 3 Solution: Start with the system: y = 3x - 7 and 5x + 2y = 19. Substitute y = 3x - 7 into the second equation: 5x + 2(3x - 7) = 19. Distribute the 2: 5x + 6x - 14 = 19.
   Full step-by-step solution

   Step 1: Start with the system: y = 3x - 7 and 5x + 2y = 19. Step 2: Substitute y = 3x - 7 into the second equation: 5x + 2(3x - 7) = 19. Step 3: Distribute the 2: 5x + 6x - 14 = 19. Step 4: Combine like terms: 11x - 14 = 19. Step 5: Add 14 to both sides: 11x = 33. Step 6: Divide both sides by 11: x = 3. The answer is 3.