Systems by Elimination

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Systems by Elimination · Grade 8 · Worksheet 3

1. Emma is planning a science club field trip to the planetarium and museum. The planetarium charges $12 per student and $15 per adult, while the museum charges $8 per student and $10 per adult. For their group of 25 people, the total cost for the planetarium would be $285, and for the museum it would be $200. How many students and how many adults are in Emma's science club? Answer: (15, 10) Solution: The elimination method for solving systems of equations involves adding or subtracting equations to eliminate one variable.
   Full step-by-step solution

   The elimination method for solving systems of equations involves adding or subtracting equations to eliminate one variable. This works because if two quantities are equal, adding them to other equal quantities maintains the equality. For example, if x + y = 10 and 2x + y = 15, subtracting the first equation from the second eliminates y, leaving x = 5. Once you find one variable, substitute back to find the other. This method is particularly useful when the coefficients of one variable are the same or can be made the same through multiplication.

2. 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 120. The price for a student ticket is $3 and for an adult ticket is $5. If the total revenue collected was $500, how many student tickets and how many adult tickets were sold? Answer: 50 student tickets and 70 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 was 120, so: s + a = 120 Write the equation for the total revenue.
   Full step-by-step solution

   Let's define variables first: 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 was 120, so: s + a = 120 Step 2: Write the equation for the total revenue. Student tickets cost $3 each, adult tickets cost $5 each, and total revenue was $500, so: 3s + 5a = 500 Step 3: Solve the system of equations. From the first equation s + a = 120, we can express s in terms of a: s = 120 - a Step 4: Substitute this expression for s into the second equation. 3(120 - a) + 5a = 500 Step 5: Simplify and solve for a. First, distribute the 3: 360 - 3a + 5a = 500 Combine like terms: 360 + 2a = 500 Subtract 360 from both sides: 2a = 500 - 360 2a = 140 Divide both sides by 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 ✓ Total revenue: 3×50 + 5×70 = 150 + 350 = 500 ✓ Therefore, 50 student tickets and 70 adult tickets were sold.

3. Liam is planning a school fundraiser and needs to determine how many student tickets and adult tickets were sold. The school sold tickets for $3 per student and $5 per adult. They sold a total of 120 tickets and collected $440. How many student tickets and how many adult tickets were sold? Answer: 80 student tickets, 40 adult tickets Solution: Let s = number of student tickets Let a = number of adult tickets 1. The total number of tickets is 120. So: s + a = 120 2.
   Full step-by-step solution

   Let's define variables for the number of tickets: Let s = number of student tickets Let a = number of adult tickets We know two facts: 1. The total number of tickets is 120. So: s + a = 120 2. The total money collected is $440. Student tickets cost $3 each, adult tickets cost $5 each. So: 3s + 5a = 440 We now have the system of equations: (1) s + a = 120 (2) 3s + 5a = 440 --- **Step 1: Solve for one variable from the first equation** From equation (1): s = 120 - a --- **Step 2: Substitute into the second equation** Replace s in equation (2) with (120 - a): 3(120 - a) + 5a = 440 --- **Step 3: Simplify and solve for a** 360 - 3a + 5a = 440 360 + 2a = 440 Subtract 360 from both sides: 2a = 80 Divide both sides by 2: a = 40 --- **Step 4: Find s** s = 120 - a = 120 - 40 = 80 --- **Step 5: Check** Number of tickets: 80 + 40 = 120 ✓ Money: 80 × 3 + 40 × 5 = 240 + 200 = 440 ✓ --- **Final answer:** 80 student tickets, 40 adult tickets

4. 7x + 2y = 27; 2x - 2y = -12 Answer: x = 3, y = 3 Solution: 7x + 2y = 27 2x - 2y = -12 (7x + 2y) + (2x - 2y) = 27 + (-12) 7x + 2x + 2y - 2y = 15 9x = 15 9x = 15 x = 15 ÷ 9 x = 5/3 Substitute x = 5/3 into the first equation: 7(5/3) + 2y = 27 35/3 + 2y = 27 2y = 27 - 35/3 2y = 81/3 - 35/3 2y = 46/3 y = 46/3 ÷ 2 y = 46/3 × 1/2 y = 46/6 y = 23/3 2(5/3) -…
   Full step-by-step solution

   Step 1: Write the original equations: 7x + 2y = 27 2x - 2y = -12 Step 2: Add the equations together to eliminate y: (7x + 2y) + (2x - 2y) = 27 + (-12) 7x + 2x + 2y - 2y = 15 9x = 15 Step 3: Solve for x: 9x = 15 x = 15 ÷ 9 x = 5/3 Step 4: Substitute x = 5/3 into the first equation: 7(5/3) + 2y = 27 35/3 + 2y = 27 Step 5: Solve for y: 2y = 27 - 35/3 2y = 81/3 - 35/3 2y = 46/3 y = 46/3 ÷ 2 y = 46/3 × 1/2 y = 46/6 y = 23/3 Step 6: Verify the solution in the second equation: 2(5/3) - 2(23/3) = 10/3 - 46/3 = -36/3 = -12 ✓ The solution is x = 5/3, y = 23/3.

5. Emma is organizing a school science fair and needs to determine how many student projects and teacher demonstrations were presented. She knows that student projects typically have 2 participants while teacher demonstrations have 1 presenter. The total number of people involved in all presentations was 85. If there were 50 presentations total, how many student projects and how many teacher demonstrations were there? Answer: (35, 15) Solution: The elimination method for solving systems of equations involves adding or subtracting equations to eliminate one variable.
   Full step-by-step solution

   The elimination method for solving systems of equations involves adding or subtracting equations to eliminate one variable. This works because if two quantities are equal, adding them to other equal quantities maintains the equality. For instance, if you had equations about apples and oranges where the total fruits and total cost were known, you could manipulate the equations to solve for each fruit type separately.

6. Liam is organizing a school fundraiser and needs to determine how many adult tickets and student tickets were sold. The total number of tickets sold was 120. Adult tickets cost $8 each, student tickets cost $5 each, and the total revenue collected was $780. How many adult tickets and how many student tickets were sold? Answer: 60 adult tickets and 60 student tickets Solution: Let A = number of adult tickets sold Let S = number of student tickets sold Write the equations from the problem.
   Full step-by-step solution

   Let’s define variables: Let A = number of adult tickets sold Let S = number of student tickets sold --- **Step 1: Write the equations from the problem.** From the total number of tickets: A + S = 120 ...(1) From the total revenue: Adult tickets cost $8 → revenue from adults = 8A Student tickets cost $5 → revenue from students = 5S Total revenue = 8A + 5S = 780 ...(2) --- **Step 2: Solve the system of equations.** From equation (1): S = 120 - A Substitute into equation (2): 8A + 5(120 - A) = 780 8A + 600 - 5A = 780 (8A - 5A) + 600 = 780 3A + 600 = 780 --- **Step 3: Solve for A.** 3A = 780 - 600 3A = 180 A = 180 / 3 A = 60 --- **Step 4: Solve for S.** S = 120 - A S = 120 - 60 S = 60 --- **Step 5: Check the solution.** Total tickets: 60 + 60 = 120 ✓ Total revenue: 8×60 + 5×60 = 480 + 300 = 780 ✓ --- **Final answer:** Adult tickets: 60 Student tickets: 60

7. A rectangular garden is drawn on a coordinate plane with corners at (2, 1), (8, 1), (8, 5), and (2, 5). A diagonal path is drawn from (2, 1) to (8, 5). What is the length of this diagonal path? Answer: 7.21 Solution: We are given two points: (2, 1) and (8, 5). We want the length of the diagonal between them. Recall the distance formula.
   Full step-by-step solution

   We are given two points: (2, 1) and (8, 5). We want the length of the diagonal between them. Step 1: Recall the distance formula. The distance d between two points (x1, y1) and (x2, y2) is: d = sqrt( (x2 - x1)^2 + (y2 - y1)^2 ) Step 2: Identify coordinates. Let (x1, y1) = (2, 1) Let (x2, y2) = (8, 5) Step 3: Calculate the differences. x2 - x1 = 8 - 2 = 6 y2 - y1 = 5 - 1 = 4 Step 4: Square the differences. (6)^2 = 36 (4)^2 = 16 Step 5: Add the squares. 36 + 16 = 52 Step 6: Take the square root. d = sqrt(52) Step 7: Simplify sqrt(52). sqrt(52) = sqrt(4 * 13) = sqrt(4) * sqrt(13) = 2 * sqrt(13) Step 8: Approximate sqrt(13). We know 3.60555^2 ≈ 13. So 2 * 3.60555 ≈ 7.2111 Step 9: Round to two decimal places. 7.2111 rounds to 7.21 Final Answer: The length of the diagonal path is 7.21.