Systems by Elimination

lessonbunny.com

Answer Key & Explanations

Systems by Elimination · Grade 8 · Worksheet 2

1. 3x + 2y = 19; 5x - 2y = 17 Answer: x = 4, y = 3.5 Solution: (3x + 2y) + (5x - 2y) = 19 + 17 3x + 5x + 2y - 2y = 36 8x = 36 x = 36 ÷ 8 x = 4.5 Substitute x = 4.5 into the first equation 3(4.5) + 2y = 19 13.5 + 2y = 19 2y = 19 - 13.5 2y = 5.5 y = 5.5 ÷ 2 y = 2.75 The solution is x = 4.5, y = 2.75.
   Full step-by-step solution

   Step 1: Add the two equations to eliminate y (3x + 2y) + (5x - 2y) = 19 + 17 3x + 5x + 2y - 2y = 36 8x = 36 Step 2: Solve for x x = 36 ÷ 8 x = 4.5 Step 3: Substitute x = 4.5 into the first equation 3(4.5) + 2y = 19 13.5 + 2y = 19 Step 4: Solve for y 2y = 19 - 13.5 2y = 5.5 y = 5.5 ÷ 2 y = 2.75 The solution is x = 4.5, y = 2.75.

2. Isabella draws a rectangle on a coordinate grid with vertices at (8, 12), (8, 20), (18, 20), and (18, 12). She then draws a diagonal from (8, 12) to (18, 20). What is the length of this diagonal? (Round to the nearest tenth if needed.) Answer: 12.8 Solution: Identify the coordinates of the diagonal endpoints: (8, 12) and (18, 20). Find the horizontal distance: 18 - 8 = 10 units. Find the vertical distance: 20 - 12 = 8 units.
   Full step-by-step solution

   Step 1: Identify the coordinates of the diagonal endpoints: (8, 12) and (18, 20). Step 2: Find the horizontal distance: 18 - 8 = 10 units. Step 3: Find the vertical distance: 20 - 12 = 8 units. Step 4: Use the Pythagorean theorem: diagonal^2 = 10^2 + 8^2 = 100 + 64 = 164. Step 5: diagonal = sqrt(164) = sqrt(4 * 41) = 2 * sqrt(41). Step 6: sqrt(41) is approximately 6.403, so 2 * 6.403 = 12.806. Step 7: Round to the nearest tenth: 12.8. The answer is 12.8.

3. 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.

4. Emma is planning a science club field trip to the planetarium. She needs to arrange transportation using vans and buses. Each van can carry 8 students and requires 1 chaperone. Each bus can carry 40 students and requires 2 chaperones. The trip has 152 students signed up and 10 chaperones available. How many vans and buses should Emma use to transport all students while using exactly the available chaperones? Answer: (4, 3) Solution: Let v = number of vans and b = number of buses.
   Full step-by-step solution

   Step 1: Let v = number of vans and b = number of buses. Step 2: Write the equation for students: 8v + 40b = 152 Step 3: Write the equation for chaperones: 1v + 2b = 10 Step 4: Multiply the chaperone equation by 8: 8v + 16b = 80 Step 5: Subtract this from the student equation: (8v + 40b) - (8v + 16b) = 152 - 80 Step 6: Simplify: 24b = 72 Step 7: Solve for b: b = 72 ÷ 24 = 3 Step 8: Substitute b = 3 into the chaperone equation: v + 2(3) = 10 Step 9: Solve for v: v + 6 = 10, so v = 4 Step 10: Check: 4 vans × 8 students = 32 students, 3 buses × 40 students = 120 students, total = 152 students. 4 vans × 1 chaperone = 4 chaperones, 3 buses × 2 chaperones = 6 chaperones, total = 10 chaperones. The answer is (4, 3).

5. Emma is planning a science fair project and needs to mix two chemical solutions. Solution A contains 15% salt and Solution B contains 40% salt. She wants to create 500 milliliters of a mixture that is 25% salt. How many milliliters of each solution should Emma use? Answer: (300,200) Solution: Let x be milliliters of Solution A (15% salt) and y be milliliters of Solution B (40% salt).
   Full step-by-step solution

   Step 1: Let x be milliliters of Solution A (15% salt) and y be milliliters of Solution B (40% salt). Step 2: Write the total volume equation: x + y = 500 Step 3: Write the salt content equation: 0.15x + 0.40y = 0.25 × 500 Step 4: Calculate the right side: 0.25 × 500 = 125, so 0.15x + 0.40y = 125 Step 5: Multiply the first equation by -0.15: -0.15x - 0.15y = -75 Step 6: Add this to the salt equation: (0.15x + 0.40y) + (-0.15x - 0.15y) = 125 + (-75) Step 7: Simplify: 0.25y = 50 Step 8: Solve for y: y = 50 ÷ 0.25 = 200 Step 9: Substitute y = 200 into x + y = 500: x + 200 = 500 Step 10: Solve for x: x = 500 - 200 = 300 Emma should use 300 ml of Solution A and 200 ml of Solution B.

6. 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), creating two right triangles. What is the area of one of these triangles? Answer: 12 Solution: (2, 1), (8, 1), (8, 5), (2, 5). - (2, 1) and (8, 1) are bottom-left and bottom-right. - (8, 5) and (2, 5) are top-right and top-left.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the rectangle and its dimensions** The rectangle has corners at: (2, 1), (8, 1), (8, 5), (2, 5). Plotting these: - (2, 1) and (8, 1) are bottom-left and bottom-right. - (8, 5) and (2, 5) are top-right and top-left. So: Length along x-axis: from x = 2 to x = 8 → length = 8 − 2 = 6 Height along y-axis: from y = 1 to y = 5 → height = 5 − 1 = 4 Area of the rectangle = length × height = 6 × 4 = 24. --- **Step 2: Diagonal from (2, 1) to (8, 5)** This diagonal splits the rectangle into two congruent right triangles. Each triangle has half the area of the rectangle. So area of one triangle = (Area of rectangle) / 2 = 24 / 2 = 12. --- **Step 3: Verification using triangle area formula** Let’s check one triangle: Take triangle with vertices (2, 1), (8, 1), (8, 5). Base = horizontal distance from (2, 1) to (8, 1) = 6 Height = vertical distance from (8, 1) to (8, 5) = 4 Area = (1/2) × base × height = (1/2) × 6 × 4 = 12. The other triangle: vertices (2, 1), (2, 5), (8, 5) Base = vertical distance from (2, 1) to (2, 5) = 4 Height = horizontal distance from (2, 5) to (8, 5) = 6 Area = (1/2) × 4 × 6 = 12. Both triangles have area 12. --- **Final answer:** 12

7. On a coordinate grid, Emma plots a rectangle with vertices at (1, 3), (1, 7), (9, 7), and (9, 3). She draws one diagonal from (1, 3) to (9, 7) and another diagonal from (1, 7) to (9, 3). The diagonals intersect at a point. What are the coordinates of the intersection point? Answer: (5, 5) Solution: The diagonals of a rectangle bisect each other, so the intersection point is the midpoint of either diagonal. Use the midpoint formula: midpoint = ((x1 + x2)/2, (y1 + y2)/2) For the diagonal from (1, 3) to (9, 7): x-coordinate: (1 + 9)/2 = 10/2 = 5 y-coordinate: (3 + 7)/2 = 10/2 = 5 The…
   Full step-by-step solution

   Step 1: The diagonals of a rectangle bisect each other, so the intersection point is the midpoint of either diagonal. Step 2: Use the midpoint formula: midpoint = ((x1 + x2)/2, (y1 + y2)/2) Step 3: For the diagonal from (1, 3) to (9, 7): x-coordinate: (1 + 9)/2 = 10/2 = 5 y-coordinate: (3 + 7)/2 = 10/2 = 5 Step 4: The intersection point is (5, 5). Step 5: Verify with the other diagonal from (1, 7) to (9, 3): x-coordinate: (1 + 9)/2 = 10/2 = 5 y-coordinate: (7 + 3)/2 = 10/2 = 5 Both diagonals give the same point, confirming the answer. The answer is (5, 5).