Systems of Equations

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Systems of Equations · Grade 8 · Worksheet 1

1. Hana is helping to organize a school book fair. She needs to order paperback books and hardcover books. Each paperback costs $11 and each hardcover costs $18. The total number of books ordered is 42, and the total cost is $588. How many paperback books and how many hardcover books did Hana order? Answer: 24 paperbacks and 18 hardcovers Solution: Let x = number of paperback books, y = number of hardcover books. Write the equation for total books: x + y = 42. Write the equation for total cost: 11x + 18y = 588.
   Full step-by-step solution

   Step 1: Let x = number of paperback books, y = number of hardcover books. Step 2: Write the equation for total books: x + y = 42. Step 3: Write the equation for total cost: 11x + 18y = 588. Step 4: Solve the first equation for x: x = 42 - y. Step 5: Substitute into the cost equation: 11(42 - y) + 18y = 588. Step 6: Distribute: 462 - 11y + 18y = 588. Step 7: Combine like terms: 462 + 7y = 588. Step 8: Subtract 462 from both sides: 7y = 126. Step 9: Divide by 7: y = 18. Step 10: Substitute back: x = 42 - 18 = 24. Hana ordered 24 paperback books and 18 hardcover books.

2. 5x + 2y = 24; 3x - 2y = 8 Answer: x = 4, y = 2 Solution: (5x + 2y) + (3x - 2y) = 24 + 8 5x + 2y + 3x - 2y = 32 8x = 32 x = 32 ÷ 8 x = 4 Substitute x = 4 into the first equation 5(4) + 2y = 24 20 + 2y = 24 2y = 24 - 20 2y = 4 y = 4 ÷ 2 y = 2 3(4) - 2(2) = 12 - 4 = 8 ✓ The solution is x = 4, y = 2.
   Full step-by-step solution

   Step 1: Add the two equations to eliminate y (5x + 2y) + (3x - 2y) = 24 + 8 5x + 2y + 3x - 2y = 32 8x = 32 Step 2: Solve for x x = 32 ÷ 8 x = 4 Step 3: Substitute x = 4 into the first equation 5(4) + 2y = 24 20 + 2y = 24 Step 4: Solve for y 2y = 24 - 20 2y = 4 y = 4 ÷ 2 y = 2 Step 5: Verify with the second equation 3(4) - 2(2) = 12 - 4 = 8 ✓ The solution is x = 4, y = 2.

3. Aroha and Tane are helping to organize a school cultural festival. They are selling two types of woven flax bags: small ones for $12 each and large ones for $18 each. At the end of the day, they sold a total of 45 bags and collected exactly $660. How many small bags and how many large bags did they sell? Answer: 25 small bags and 20 large bags Solution: Let s = number of small bags, l = number of large bags.
   Full step-by-step solution

   Let s = number of small bags, l = number of large bags. Equation 1 (total bags): s + l = 45 Equation 2 (total money): 12s + 18l = 660 Solve equation 1 for s: s = 45 - l Substitute into equation 2: 12(45 - l) + 18l = 660 Simplify: 540 - 12l + 18l = 660 Combine like terms: 540 + 6l = 660 Subtract 540 from both sides: 6l = 120 Divide both sides by 6: l = 20 Now find s: s = 45 - 20 = 25 Check: 12(25) + 18(20) = 300 + 360 = 660. Correct. They sold 25 small bags and 20 large bags.

4. 2x + 3y = 19; 4x - y = 17 Answer: x = 5, y = 3 Solution: Start with the system: 2x + 3y = 19 and 4x - y = 17 Multiply the second equation by 3 to eliminate y: 3(4x - y) = 3(17) → 12x - 3y = 51 Add this to the first equation: (2x + 3y) + (12x - 3y) = 19 + 51 → 14x = 70 Solve for x: x = 70 ÷ 14 = 5 Substitute x = 5 into the second equation: 4(5) - y =…
   Full step-by-step solution

   Step 1: Start with the system: 2x + 3y = 19 and 4x - y = 17 Step 2: Multiply the second equation by 3 to eliminate y: 3(4x - y) = 3(17) → 12x - 3y = 51 Step 3: Add this to the first equation: (2x + 3y) + (12x - 3y) = 19 + 51 → 14x = 70 Step 4: Solve for x: x = 70 ÷ 14 = 5 Step 5: Substitute x = 5 into the second equation: 4(5) - y = 17 → 20 - y = 17 Step 6: Solve for y: -y = 17 - 20 = -3 → y = 3 Step 7: Check in first equation: 2(5) + 3(3) = 10 + 9 = 19 ✓ The solution is x = 5, y = 3.

5. 2x + 3y = 12, x - y = 1 Answer: x = 3, y = 2 Solution: (1) 2x + 3y = 12 (2) x - y = 1 Solve one equation for one variable. From equation (2), x - y = 1, we can solve for x: x = y + 1 Substitute this expression for x into equation (1).
   Full step-by-step solution

   We are given the system of equations: (1) 2x + 3y = 12 (2) x - y = 1 Step 1: Solve one equation for one variable. From equation (2), x - y = 1, we can solve for x: x = y + 1 Step 2: Substitute this expression for x into equation (1). Replace x in equation (1) with (y + 1): 2(y + 1) + 3y = 12 Step 3: Simplify and solve for y. 2y + 2 + 3y = 12 5y + 2 = 12 Subtract 2 from both sides: 5y = 10 Divide both sides by 5: y = 2 Step 4: Substitute y = 2 back into x = y + 1. x = 2 + 1 x = 3 Step 5: Check the solution in both original equations. For equation (1): 2(3) + 3(2) = 6 + 6 = 12 ✓ For equation (2): 3 - 2 = 1 ✓ Final answer: x = 3, y = 2

6. Liam is planning a school fundraiser and needs to order pizza for the event. He knows that cheese pizzas cost $12 each and pepperoni pizzas cost $15 each. His total budget for pizza is $300, and he needs to order exactly 22 pizzas total. How many cheese pizzas and how many pepperoni pizzas should Liam order to spend exactly his budget? Answer: 10 cheese pizzas and 12 pepperoni pizzas Solution: Let c = number of cheese pizzas Let p = number of pepperoni pizzas 1. The total number of pizzas is 22: c + p = 22 2.
   Full step-by-step solution

   Let's define variables first. Let c = number of cheese pizzas Let p = number of pepperoni pizzas We know: 1. The total number of pizzas is 22: c + p = 22 2. The total cost is $300: 12c + 15p = 300 --- **Step 1: Solve for one variable from the first equation** From c + p = 22, we get: c = 22 - p --- **Step 2: Substitute into the second equation** 12(22 - p) + 15p = 300 --- **Step 3: Simplify and solve for p** 264 - 12p + 15p = 300 264 + 3p = 300 3p = 300 - 264 3p = 36 p = 12 --- **Step 4: Find c** c = 22 - p = 22 - 12 = 10 --- **Step 5: Check the cost** Cheese pizzas: 10 × 12 = 120 Pepperoni pizzas: 12 × 15 = 180 Total = 120 + 180 = 300 ✓ Total pizzas = 10 + 12 = 22 ✓ --- **Final answer:** Liam should order 10 cheese pizzas and 12 pepperoni pizzas.

7. A triangular park is drawn on a coordinate plane with vertices at A(7, 10), B(19, 10), and C(13, 22). A straight walking path runs from vertex A to the midpoint of side BC. Another straight path runs from vertex B to the midpoint of side AC. Find the coordinates of the point where these two paths intersect. Answer: (13, 14) Solution: Find the midpoint of side BC. B(19, 10) and C(13, 22). Midpoint M_BC = ((19+13)/2, (10+22)/2) = (32/2, 32/2) = (16, 16).
   Full step-by-step solution

   Step 1: Find the midpoint of side BC. B(19, 10) and C(13, 22). Midpoint M_BC = ((19+13)/2, (10+22)/2) = (32/2, 32/2) = (16, 16). Step 2: Find the midpoint of side AC. A(7, 10) and C(13, 22). Midpoint M_AC = ((7+13)/2, (10+22)/2) = (20/2, 32/2) = (10, 16). Step 3: Find the equation of the line from A(7, 10) to M_BC(16, 16). Slope = (16-10)/(16-7) = 6/9 = 2/3. Using point-slope form with A(7,10): y - 10 = (2/3)(x - 7). Multiply by 3: 3y - 30 = 2x - 14. Rearrange: 3y = 2x + 16, so y = (2/3)x + 16/3. Step 4: Find the equation of the line from B(19, 10) to M_AC(10, 16). Slope = (16-10)/(10-19) = 6/(-9) = -2/3. Using point-slope form with B(19,10): y - 10 = (-2/3)(x - 19). Multiply by 3: 3y - 30 = -2x + 38. Rearrange: 3y = -2x + 68, so y = (-2/3)x + 68/3. Step 5: Solve the system of equations. Set (2/3)x + 16/3 = (-2/3)x + 68/3. Multiply by 3: 2x + 16 = -2x + 68. Add 2x to both sides: 4x + 16 = 68. Subtract 16: 4x = 52. Divide by 4: x = 13. Step 6: Substitute x = 13 into either equation. Using y = (2/3)(13) + 16/3 = 26/3 + 16/3 = 42/3 = 14. The intersection point is (13, 14).

8. Aroha is helping organize the school's recycling drive. She needs to collect a total of 45 kilograms of paper and cardboard. Each bag of paper weighs 3 kilograms, and each bag of cardboard weighs 5 kilograms. If she collects a total of 11 bags, how many bags of paper and how many bags of cardboard does she collect? Answer: 5 bags of paper and 6 bags of cardboard Solution: Let x = number of bags of paper and y = number of bags of cardboard.
   Full step-by-step solution

   Step 1: Let x = number of bags of paper and y = number of bags of cardboard. Step 2: Equation for total bags: x + y = 11 Step 3: Equation for total weight: 3x + 5y = 45 Step 4: Solve the first equation for x: x = 11 - y Step 5: Substitute into the second equation: 3(11 - y) + 5y = 45 Step 6: Simplify: 33 - 3y + 5y = 45 Step 7: Combine like terms: 33 + 2y = 45 Step 8: Subtract 33 from both sides: 2y = 12 Step 9: Divide both sides by 2: y = 6 Step 10: Substitute y = 6 into x = 11 - y: x = 11 - 6 = 5 Step 11: Aroha collects 5 bags of paper and 6 bags of cardboard.