Systems of Equations

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Systems of Equations · Grade 8 · Worksheet 2

1. A rectangular garden is drawn on a coordinate plane. The garden's corners are at points (2, 1), (8, 1), (8, 5), and (2, 5). A diagonal path is drawn from corner (2, 1) to corner (8, 5). Find the point where this diagonal path intersects a vertical line drawn through x = 5. What are the coordinates of this intersection point? Answer: (5, 3) Solution: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) The diagonal goes from A = (2, 1) to C = (8, 5). We want the point where this diagonal intersects the vertical line x = 5.
   Full step-by-step solution

   Let's solve this step-by-step. --- **Step 1: Understand the problem** We have a rectangle with corners: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) The diagonal goes from A = (2, 1) to C = (8, 5). We want the point where this diagonal intersects the vertical line x = 5. --- **Step 2: Find the equation of the diagonal AC** Points A(2, 1) and C(8, 5): Slope m = (change in y) / (change in x) m = (5 - 1) / (8 - 2) = 4 / 6 = 2/3 Equation of line through A(2, 1) with slope 2/3: y - 1 = (2/3)(x - 2) Simplify: y - 1 = (2/3)x - 4/3 y = (2/3)x - 4/3 + 1 y = (2/3)x - 4/3 + 3/3 y = (2/3)x - 1/3 So the diagonal line is: y = (2/3)x - 1/3 --- **Step 3: Find intersection with x = 5** Substitute x = 5 into the equation: y = (2/3)(5) - 1/3 y = 10/3 - 1/3 y = 9/3 y = 3 --- **Step 4: Conclusion** The intersection point is (5, 3). --- **Final answer:** (5, 3)

2. 3x + 2y = 16; 2x - y = 6 Answer: x = 4, y = 2 Solution: Start with the system: 3x + 2y = 16 and 2x - y = 6 Multiply the second equation by 2 to eliminate y: 2(2x - y) = 2(6) → 4x - 2y = 12 Add this to the first equation: (3x + 2y) + (4x - 2y) = 16 + 12 → 7x = 28 Solve for x: x = 28 ÷ 7 = 4 Substitute x = 4 into the second equation: 2(4) - y = 6 → 8 -…
   Full step-by-step solution

   Step 1: Start with the system: 3x + 2y = 16 and 2x - y = 6 Step 2: Multiply the second equation by 2 to eliminate y: 2(2x - y) = 2(6) → 4x - 2y = 12 Step 3: Add this to the first equation: (3x + 2y) + (4x - 2y) = 16 + 12 → 7x = 28 Step 4: Solve for x: x = 28 ÷ 7 = 4 Step 5: Substitute x = 4 into the second equation: 2(4) - y = 6 → 8 - y = 6 Step 6: Solve for y: -y = 6 - 8 → -y = -2 → y = 2 Step 7: Check in first equation: 3(4) + 2(2) = 12 + 4 = 16 ✓ The solution is x = 4, y = 2.

3. 3x + 2y = 16; 2x - y = 1 Answer: x = 2, y = 3 Solution: Start with the system: 3x + 2y = 16 and 2x - y = 1 Multiply the second equation by 2 to match the y-coefficient: 2(2x - y) = 2(1) → 4x - 2y = 2 Add this to the first equation: (3x + 2y) + (4x - 2y) = 16 + 2 → 7x = 18 Solve for x: x = 18/7 = 2 Substitute x = 2 into the second equation: 2(2) - y =…
   Full step-by-step solution

   Step 1: Start with the system: 3x + 2y = 16 and 2x - y = 1 Step 2: Multiply the second equation by 2 to match the y-coefficient: 2(2x - y) = 2(1) → 4x - 2y = 2 Step 3: Add this to the first equation: (3x + 2y) + (4x - 2y) = 16 + 2 → 7x = 18 Step 4: Solve for x: x = 18/7 = 2 Step 5: Substitute x = 2 into the second equation: 2(2) - y = 1 → 4 - y = 1 Step 6: Solve for y: -y = 1 - 4 → -y = -3 → y = 3 Step 7: Check in first equation: 3(2) + 2(3) = 6 + 6 = 12 The solution is x = 2, y = 3.

4. Olivia is organizing a school bake sale to raise money for new library books. She decides to sell two types of baked goods: cupcakes and brownies. Each cupcake costs $3 to make, and each brownie costs $5 to make. Olivia has a total budget of $99 for ingredients. She also knows that she needs to make a total of 25 baked goods to have enough for the sale. How many cupcakes and how many brownies should Olivia make to use exactly her budget and meet the total number of baked goods? Answer: 13 cupcakes and 12 brownies Solution: Define variables. Let x = number of cupcakes and y = number of brownies. Write the equation for the total number of baked goods: x + y = 25.
   Full step-by-step solution

   Step 1: Define variables. Let x = number of cupcakes and y = number of brownies. Step 2: Write the equation for the total number of baked goods: x + y = 25. Step 3: Write the equation for the total cost: 3x + 5y = 99. Step 4: Solve the first equation for x: x = 25 - y. Step 5: Substitute this expression for x into the second equation: 3(25 - y) + 5y = 99. Step 6: Distribute the 3: 75 - 3y + 5y = 99. Step 7: Combine like terms: 75 + 2y = 99. Step 8: Subtract 75 from both sides: 2y = 24. Step 9: Divide both sides by 2: y = 12. Step 10: Substitute y = 12 back into x = 25 - y: x = 25 - 12 = 13. The answer is 13 cupcakes and 12 brownies.

5. A rectangle is drawn on a coordinate plane with vertices at (2, 1), (6, 1), (6, 4), and (2, 4). A line passes through the points (0, 5) and (8, -3). Find the coordinates of the point where this line intersects the rectangle's perimeter. Answer: (3, 2) Solution: To solve problems involving lines intersecting geometric shapes on a coordinate plane, we use the equation of the line and check for intersections with each boundary of the shape.
   Full step-by-step solution

   To solve problems involving lines intersecting geometric shapes on a coordinate plane, we use the equation of the line and check for intersections with each boundary of the shape. The line equation can be found using the slope formula and point-slope form. For rectangles, we check intersections with the horizontal lines (top and bottom) and vertical lines (left and right sides) that define the shape's boundaries. The valid intersection point will be the one that lies within the segment boundaries of both the line and the rectangle side.

6. Hana is helping to organize the school athletics day. She needs to buy two types of drinks: sports drinks and water bottles. Each sports drink costs $4 and each water bottle costs $2. Hana buys a total of 50 drinks and spends exactly $160. How many sports drinks and how many water bottles does Hana buy? Answer: 30 sports drinks and 20 water bottles Solution: Let x = number of sports drinks and y = number of water bottles.
   Full step-by-step solution

   Step 1: Let x = number of sports drinks and y = number of water bottles. Step 2: Total number of drinks: x + y = 50 Step 3: Total cost: 4x + 2y = 160 Step 4: Solve the first equation for x: x = 50 - y Step 5: Substitute into the second equation: 4(50 - y) + 2y = 160 Step 6: Simplify: 200 - 4y + 2y = 160 Step 7: Combine like terms: 200 - 2y = 160 Step 8: Subtract 200 from both sides: -2y = -40 Step 9: Divide both sides by -2: y = 20 Step 10: Substitute y = 20 into x = 50 - y: x = 50 - 20 = 30 Hana buys 30 sports drinks and 20 water bottles.

7. 5x + 2y = 40; 3x - 2y = 0. Solve for x and y. Answer: x = 5, y = 7.5 Solution: Write the system: 5x + 2y = 40 and 3x - 2y = 0. Add the two equations to eliminate y: (5x + 2y) + (3x - 2y) = 40 + 0 → 8x = 40. Solve for x: x = 40 ÷ 8 = 5.
   Full step-by-step solution

   Step 1: Write the system: 5x + 2y = 40 and 3x - 2y = 0. Step 2: Add the two equations to eliminate y: (5x + 2y) + (3x - 2y) = 40 + 0 → 8x = 40. Step 3: Solve for x: x = 40 ÷ 8 = 5. Step 4: Substitute x = 5 into the first equation: 5(5) + 2y = 40 → 25 + 2y = 40. Step 5: Solve for y: 2y = 40 - 25 → 2y = 15 → y = 7.5. Step 6: Check in the second equation: 3(5) - 2(7.5) = 15 - 15 = 0 ✓. The solution is x = 5, y = 7.5.

8. 4x + 5y = 47; 3x - 2y = 18. Solve for x and y. Answer: x = 8, y = 3 Solution: Start with the system: 4x + 5y = 47 and 3x - 2y = 18. Multiply the first equation by 2: 2(4x + 5y) = 2(47) → 8x + 10y = 94. Multiply the second equation by 5: 5(3x - 2y) = 5(18) → 15x - 10y = 90.
   Full step-by-step solution

   Step 1: Start with the system: 4x + 5y = 47 and 3x - 2y = 18. Step 2: Multiply the first equation by 2: 2(4x + 5y) = 2(47) → 8x + 10y = 94. Step 3: Multiply the second equation by 5: 5(3x - 2y) = 5(18) → 15x - 10y = 90. Step 4: Add the two equations: (8x + 10y) + (15x - 10y) = 94 + 90 → 23x = 184. Step 5: Solve for x: x = 184 ÷ 23 = 8. Step 6: Substitute x = 8 into the first equation: 4(8) + 5y = 47 → 32 + 5y = 47. Step 7: Solve for y: 5y = 47 - 32 → 5y = 15 → y = 3. Step 8: Check in the second equation: 3(8) - 2(3) = 24 - 6 = 18 ✓. The solution is x = 8, y = 3.