Worksheet 1Worksheet 2Worksheet 3

The download includes this worksheet followed by the answer key with explanations. In the print dialog choose “Save as PDF” to download.

lessonbunny.com
Name: ______________________________ Date: ______________

Systems by Substitution

Grade 8 · Algebra · Worksheet 2

  1. A rectangular garden has a length that is 5 meters more than twice its width. The perimeter of the garden is 46 meters. Using the substitution method, find the dimensions of the garden by setting up a system of equations where l represents the length and w represents the width. Answer: ______________
  2. 2x + 5y = 35; y = 3x - 10; x = ? Answer: ______________
  3. 2x + 3y = 12 and y = x - 1, find x and y Answer: ______________
  4. Tane is designing a triangular skateboard ramp on a coordinate grid. The ramp has vertices at A(10, 2), B(22, 2), and C(13, 14). A straight support beam runs from vertex C to the midpoint of side AB. The equation for side AB is y = 2. Using the substitution method, find the coordinates of the point where the support beam intersects side AB. Answer: ______________
  5. Liam is planning a school fundraiser and needs to determine how many student tickets and adult tickets were sold. The total number of tickets sold was 85. Student tickets cost $3 each, adult tickets cost $5 each, and the total amount collected was $335. How many student tickets and how many adult tickets were sold? Answer: ______________
  6. Liam is planning a school fundraiser and needs to figure out how many student tickets and adult tickets were sold. The total number of tickets sold was 120. Student tickets cost $3 each, adult tickets cost $5 each, and the total amount of money collected was $500. How many student tickets and how many adult tickets were sold? Answer: ______________
  7. 2x + y = 11; y = 3x - 4; x = ? Answer: ______________
  8. 3x - 2y = 8; y = 2x - 5; x = ? Answer: ______________
lessonbunny.com

Answer Key & Explanations

Systems by Substitution · Grade 8 · Worksheet 2

  1. A rectangular garden has a length that is 5 meters more than twice its width. The perimeter of the garden is 46 meters. Using the substitution method, find the dimensions of the garden by setting up a system of equations where l represents the length and w represents the width. Answer: length = 17 meters, width = 6 meters Solution: Define variables and write equations from the problem. - l = length (in meters) - w = width (in meters) 1. "length is 5 meters more than twice its width" means: l = 2w + 5 2.
    Full step-by-step solution

    Let's solve step by step. --- **Step 1: Define variables and write equations from the problem.** Let: - l = length (in meters) - w = width (in meters) From the problem: 1. "length is 5 meters more than twice its width" means: l = 2w + 5 2. "perimeter is 46 meters" means: Perimeter = 2l + 2w = 46 So the system of equations is: Equation (1): l = 2w + 5 Equation (2): 2l + 2w = 46 --- **Step 2: Substitute Equation (1) into Equation (2).** From Equation (1), l = 2w + 5. Substitute into Equation (2): 2(2w + 5) + 2w = 46 --- **Step 3: Simplify and solve for w.** First, distribute: 4w + 10 + 2w = 46 Combine like terms: 6w + 10 = 46 Subtract 10 from both sides: 6w = 36 Divide both sides by 6: w = 6 --- **Step 4: Solve for l using w = 6.** From Equation (1): l = 2w + 5 l = 2(6) + 5 l = 12 + 5 l = 17 --- **Step 5: State the dimensions.** Width = 6 meters Length = 17 meters --- **Step 6: Check the solution.** Perimeter = 2(17) + 2(6) = 34 + 12 = 46 meters ✓ Length = 17, which is 5 more than twice the width: twice the width = 12, plus 5 = 17 ✓ --- Final answer: length = 17 meters, width = 6 meters

  2. 2x + 5y = 35; y = 3x - 10; x = ? Answer: 5 Solution: Start with the system: 2x + 5y = 35 and y = 3x - 10. Substitute y = 3x - 10 into the first equation: 2x + 5(3x - 10) = 35. Distribute the 5: 2x + 15x - 50 = 35.
    Full step-by-step solution

    Step 1: Start with the system: 2x + 5y = 35 and y = 3x - 10. Step 2: Substitute y = 3x - 10 into the first equation: 2x + 5(3x - 10) = 35. Step 3: Distribute the 5: 2x + 15x - 50 = 35. Step 4: Combine like terms: 17x - 50 = 35. Step 5: Add 50 to both sides: 17x = 85. Step 6: Divide both sides by 17: x = 5. The answer is 5.

  3. 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

  4. Tane is designing a triangular skateboard ramp on a coordinate grid. The ramp has vertices at A(10, 2), B(22, 2), and C(13, 14). A straight support beam runs from vertex C to the midpoint of side AB. The equation for side AB is y = 2. Using the substitution method, find the coordinates of the point where the support beam intersects side AB. Answer: (16, 2) Solution: Find the midpoint of side AB. A is at (10, 2) and B is at (22, 2).
    Full step-by-step solution

    Step 1: Find the midpoint of side AB. A is at (10, 2) and B is at (22, 2). Using the midpoint formula ((x1 + x2)/2, (y1 + y2)/2): x-coordinate = (10 + 22)/2 = 32/2 = 16 y-coordinate = (2 + 2)/2 = 4/2 = 2 So the midpoint is at (16, 2). Step 2: The support beam runs from vertex C(13, 14) to the midpoint (16, 2). Since this is a straight line, the point where it meets side AB is exactly the midpoint we found. Step 3: We are given that side AB has equation y = 2. The support beam's intersection with side AB is at the midpoint, which has y-coordinate 2 (satisfying the equation y = 2). Step 4: The coordinates are (16, 2). The answer is (16, 2).

  5. Liam is planning a school fundraiser and needs to determine how many student tickets and adult tickets were sold. The total number of tickets sold was 85. Student tickets cost $3 each, adult tickets cost $5 each, and the total amount collected was $335. How many student tickets and how many adult tickets were sold? Answer: 45 student tickets and 40 adult tickets Solution: Let s = number of student tickets Let a = number of adult tickets Write the equation for the total number of tickets. We know the total tickets sold is 85. s + a = 85 Write the equation for the total money collected.
    Full step-by-step solution

    Let's define variables: Let s = number of student tickets Let a = number of adult tickets Step 1: Write the equation for the total number of tickets. We know the total tickets sold is 85. So: s + a = 85 Step 2: Write the equation for the total money collected. Student tickets cost $3 each, so money from students = 3s Adult tickets cost $5 each, so money from adults = 5a Total money = $335 So: 3s + 5a = 335 Step 3: Solve the system of equations. From s + a = 85, we get s = 85 - a Step 4: Substitute into the money equation. 3(85 - a) + 5a = 335 Step 5: Simplify and solve for a. 255 - 3a + 5a = 335 255 + 2a = 335 2a = 335 - 255 2a = 80 a = 40 Step 6: Find s. s = 85 - a = 85 - 40 = 45 Step 7: Check the solution. Number of tickets: 45 + 40 = 85 ✓ Money: 45 * 3 + 40 * 5 = 135 + 200 = 335 ✓ ANSWER: 45 student tickets and 40 adult tickets.

  6. Liam is planning a school fundraiser and needs to figure out how many student tickets and adult tickets were sold. The total number of tickets sold was 120. Student tickets cost $3 each, adult tickets cost $5 each, and the total amount of money collected was $500. How many student tickets and how many adult tickets were sold? Answer: 50 student tickets and 70 adult tickets Solution: s = number of student tickets a = number of adult tickets Write the equation for the total number of tickets. We know the total tickets sold is 120. So: s + a = 120 Write the equation for the total money collected.
    Full step-by-step solution

    Let's define variables first. Let: s = number of student tickets a = number of adult tickets Step 1: Write the equation for the total number of tickets. We know the total tickets sold is 120. So: s + a = 120 Step 2: Write the equation for the total money collected. Student tickets cost $3 each, so money from students = 3s Adult tickets cost $5 each, so money from adults = 5a Total money = $500 So: 3s + 5a = 500 Step 3: Solve the system of equations. We have: (1) s + a = 120 (2) 3s + 5a = 500 From equation (1), we can write: s = 120 - a Step 4: Substitute s into equation (2). Replace s in equation (2) with (120 - a): 3(120 - a) + 5a = 500 Step 5: Simplify and solve for a. 360 - 3a + 5a = 500 360 + 2a = 500 2a = 500 - 360 2a = 140 a = 140 / 2 a = 70 Step 6: Find s using s = 120 - a. s = 120 - 70 s = 50 Step 7: Check the solution. Total tickets: 50 + 70 = 120 (correct) Total money: 3*50 + 5*70 = 150 + 350 = 500 (correct) Final answer: 50 student tickets and 70 adult tickets were sold.

  7. 2x + y = 11; y = 3x - 4; x = ? Answer: 3 Solution: 1) 2x + y = 11 2) y = 3x - 4 Since equation 2 gives y in terms of x, we can substitute this expression for y into equation 1. Substitute y from equation 2 into equation 1.
    Full step-by-step solution

    We are given the system of equations: 1) 2x + y = 11 2) y = 3x - 4 Since equation 2 gives y in terms of x, we can substitute this expression for y into equation 1. Step 1: Substitute y from equation 2 into equation 1. Replace y in equation 1 with (3x - 4): 2x + (3x - 4) = 11 Step 2: Combine like terms. 2x + 3x - 4 = 11 5x - 4 = 11 Step 3: Add 4 to both sides to isolate the term with x. 5x - 4 + 4 = 11 + 4 5x = 15 Step 4: Divide both sides by 5 to solve for x. 5x / 5 = 15 / 5 x = 3 Thus, the solution is x = 3. We can check by substituting x = 3 into equation 2: y = 3(3) - 4 = 9 - 4 = 5 Then equation 1: 2(3) + 5 = 6 + 5 = 11, which is correct. Final answer: x = 3

  8. 3x - 2y = 8; y = 2x - 5; x = ? Answer: 2 Solution: Start with the equations: 3x - 2y = 8 and y = 2x - 5. Substitute (2x - 5) for y in the first equation: 3x - 2(2x - 5) = 8. Distribute the -2: 3x - 4x + 10 = 8.
    Full step-by-step solution

    Step 1: Start with the equations: 3x - 2y = 8 and y = 2x - 5. Step 2: Substitute (2x - 5) for y in the first equation: 3x - 2(2x - 5) = 8. Step 3: Distribute the -2: 3x - 4x + 10 = 8. Step 4: Combine like terms: -x + 10 = 8. Step 5: Subtract 10 from both sides: -x = -2. Step 6: Multiply both sides by -1: x = 2. The value of x is 2.