Systems by Substitution

lessonbunny.com

Answer Key & Explanations

Systems by Substitution · Grade 8 · Worksheet 1

1. A rectangular garden has a length that is 3 meters more than twice its width. The perimeter of the garden is 36 meters. Using the substitution method, find the dimensions of the garden. Answer: width = 5 meters, length = 13 meters Solution: Let width = w meters Let length = l meters "length is 3 meters more than twice its width" means: l = 2w + 3 Perimeter = 2 × (length + width) Given perimeter = 36 meters, so: 2(l + w) = 36 Substitute the expression for l into the perimeter equation From l = 2w + 3, substitute into 2(l + w) = 36:…
   Full step-by-step solution

   Let's solve step-by-step using the substitution method. --- **Step 1: Define variables** Let width = w meters Let length = l meters --- **Step 2: Translate the problem into equations** From the problem: "length is 3 meters more than twice its width" means: l = 2w + 3 Also, perimeter of a rectangle is: Perimeter = 2 × (length + width) Given perimeter = 36 meters, so: 2(l + w) = 36 --- **Step 3: Substitute the expression for l into the perimeter equation** From l = 2w + 3, substitute into 2(l + w) = 36: 2( (2w + 3) + w ) = 36 --- **Step 4: Simplify and solve for w** Inside the parentheses: 2w + 3 + w = 3w + 3 So: 2(3w + 3) = 36 Divide both sides by 2: 3w + 3 = 18 Subtract 3 from both sides: 3w = 15 Divide by 3: w = 5 --- **Step 5: Find length l** l = 2w + 3 l = 2(5) + 3 l = 10 + 3 l = 13 --- **Step 6: Verify** Perimeter = 2 × (l + w) = 2 × (13 + 5) = 2 × 18 = 36 ✓ Length (13) is 3 more than twice width (2×5=10, 10+3=13) ✓ --- **Final answer:** width = 5 meters, length = 13 meters

2. 2x + 3y = 12 and y = x - 1 Answer: x = 3, y = 2 Solution: 1) 2x + 3y = 12 2) y = x - 1 Substitute 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 have the system of equations: 1) 2x + 3y = 12 2) y = x - 1 Step 1: Substitute 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 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: 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. Equation 1: 2x + 3y = 12 Substitute x = 3, y = 2: 2(3) + 3(2) = 6 + 6 = 12 This matches the given equation. Final answer: x = 3, y = 2

3. Hana is helping to organize the school library. She needs to sort new books onto shelves. She has a total of 144 books to place on two types of shelves: small shelves that hold 12 books each and large shelves that hold 24 books each. She uses a total of 9 shelves. How many small shelves and how many large shelves does Hana use? Answer: x = 6, y = 3 Solution: Let x be the number of small shelves and y be the number of large shelves. Equation 1 (total shelves): x + y = 9 Equation 2 (total books): 12x + 24y = 144 Solve Equation 1 for x: x = 9 - y Substitute into Equation 2: 12(9 - y) + 24y = 144 Simplify: 108 - 12y + 24y = 144 Combine y terms: 108 +…
   Full step-by-step solution

   Let x be the number of small shelves and y be the number of large shelves. Equation 1 (total shelves): x + y = 9 Equation 2 (total books): 12x + 24y = 144 Solve Equation 1 for x: x = 9 - y Substitute into Equation 2: 12(9 - y) + 24y = 144 Simplify: 108 - 12y + 24y = 144 Combine y terms: 108 + 12y = 144 Subtract 108 from both sides: 12y = 36 Divide both sides by 12: y = 3 Substitute y = 3 into x = 9 - y: x = 9 - 3 = 6 Hana uses 6 small shelves and 3 large shelves.

4. Charlotte is helping to organize a school fundraiser. She is selling two types of gift cards: coffee shop cards worth $7 each and bookstore cards worth $12 each. She sold a total of 27 gift cards and raised $249. How many coffee shop cards and how many bookstore cards did Charlotte sell? Answer: 15 coffee shop cards and 12 bookstore cards Solution: Let c = number of coffee shop cards and b = number of bookstore cards.
   Full step-by-step solution

   Let c = number of coffee shop cards and b = number of bookstore cards. Equation 1 (total cards): c + b = 27 Equation 2 (total money): 7c + 12b = 249 Solve Equation 1 for c: c = 27 - b Substitute into Equation 2: 7(27 - b) + 12b = 249 Simplify: 189 - 7b + 12b = 249 189 + 5b = 249 5b = 60 b = 12 Now substitute b = 12 into c = 27 - b: c = 27 - 12 = 15 Check: 7(15) + 12(12) = 105 + 144 = 249 dollars, and 15 + 12 = 27 cards. Final answer: Charlotte sold 15 coffee shop cards and 12 bookstore cards.

5. x = 2y - 7; 3x + 5y = 34; x = ? Answer: 3 Solution: Start with the system: x = 2y - 7 and 3x + 5y = 34. Substitute (2y - 7) for x in the second equation: 3(2y - 7) + 5y = 34. Distribute the 3: 6y - 21 + 5y = 34.
   Full step-by-step solution

   Step 1: Start with the system: x = 2y - 7 and 3x + 5y = 34. Step 2: Substitute (2y - 7) for x in the second equation: 3(2y - 7) + 5y = 34. Step 3: Distribute the 3: 6y - 21 + 5y = 34. Step 4: Combine like terms: 11y - 21 = 34. Step 5: Add 21 to both sides: 11y = 55. Step 6: Divide both sides by 11: y = 5. Step 7: Substitute y = 5 into the first equation: x = 2(5) - 7 = 10 - 7 = 3. The value of x is 3.

6. Emma is organizing a school bake sale. She needs to buy two types of flour: white flour costs $5 per bag, and whole wheat flour costs $8 per bag. She has a total budget of $115 and wants to buy exactly 20 bags of flour in total. How many bags of white flour and how many bags of whole wheat flour should Emma buy? Answer: 15 bags of white flour and 5 bags of whole wheat flour Solution: Let w = number of bags of white flour and h = number of bags of whole wheat flour.
   Full step-by-step solution

   Let w = number of bags of white flour and h = number of bags of whole wheat flour. Equation 1 (total bags): w + h = 20 Equation 2 (total cost): 5w + 8h = 115 Solve Equation 1 for w: w = 20 - h Substitute into Equation 2: 5(20 - h) + 8h = 115 Simplify: 100 - 5h + 8h = 115 100 + 3h = 115 3h = 15 h = 5 Now substitute h = 5 into w = 20 - h: w = 20 - 5 = 15 Check: 5(15) + 8(5) = 75 + 40 = 115, and 15 + 5 = 20 bags. Emma should buy 15 bags of white flour and 5 bags of whole wheat flour.

7. 4x + 3y = 29; y = 2x - 7; x = ? Answer: 5 Solution: Start with the system: 4x + 3y = 29 and y = 2x - 7. Substitute y = 2x - 7 into the first equation: 4x + 3(2x - 7) = 29. Distribute the 3: 4x + 6x - 21 = 29.
   Full step-by-step solution

   Step 1: Start with the system: 4x + 3y = 29 and y = 2x - 7. Step 2: Substitute y = 2x - 7 into the first equation: 4x + 3(2x - 7) = 29. Step 3: Distribute the 3: 4x + 6x - 21 = 29. Step 4: Combine like terms: 10x - 21 = 29. Step 5: Add 21 to both sides: 10x = 50. Step 6: Divide both sides by 10: x = 5. The value of x is 5.

8. 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).