Systems Word Problems

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Answer Key & Explanations

Systems Word Problems · Grade 8 · Worksheet 2

1. Aroha and Tane are selling tickets for a school play. Adult tickets cost $9 each and student tickets cost $5 each. They sold 117 tickets in total and collected $789. How many adult tickets and how many student tickets were sold? Answer: Adult tickets: 51, Student tickets: 66 Solution: Let a = number of adult tickets, s = number of student tickets. Equation 1 (total tickets): a + s = 117 Equation 2 (total money): 9a + 5s = 789 Solve by substitution: From equation 1, s = 117 - a.
   Full step-by-step solution

   Let a = number of adult tickets, s = number of student tickets. Equation 1 (total tickets): a + s = 117 Equation 2 (total money): 9a + 5s = 789 Solve by substitution: From equation 1, s = 117 - a. Substitute into equation 2: 9a + 5(117 - a) = 789 9a + 585 - 5a = 789 4a + 585 = 789 4a = 204 a = 51 Then s = 117 - 51 = 66. Check: 9(51) + 5(66) = 459 + 330 = 789. Correct. Answer: 51 adult tickets and 66 student tickets.

2. Isabella and Mason together have 72 comic books. If Isabella gives 7 comic books to Mason, she will have twice as many comic books as Mason. How many comic books does each person have originally? Answer: Isabella has 57 comic books, Mason has 15 comic books Solution: Let i = number of comic books Isabella has originally, and m = number of comic books Mason has originally.
   Full step-by-step solution

   Let i = number of comic books Isabella has originally, and m = number of comic books Mason has originally. Equation 1 (total): i + m = 72 Equation 2 (after transfer): After Isabella gives 7 comic books to Mason, Isabella has i - 7 and Mason has m + 7. At that point, Isabella has twice as many as Mason: i - 7 = 2(m + 7) Simplify Equation 2: i - 7 = 2m + 14 → i = 2m + 21 Substitute into Equation 1: (2m + 21) + m = 72 → 3m + 21 = 72 → 3m = 51 → m = 17 Then i = 2(17) + 21 = 34 + 21 = 55 Check: Isabella gives 7 to Mason: Isabella has 48, Mason has 24. 48 is twice 24. ✓ So Isabella originally has 55 comic books and Mason has 17 comic books.

3. 2x + 3y = 16 and 4x - y = 10, find x and y Answer: x = 3.5, y = 4 Solution: Solve the second equation for y: 4x - y = 10 → -y = 10 - 4x → y = 4x - 10 Substitute y = 4x - 10 into the first equation: 2x + 3(4x - 10) = 16 Simplify: 2x + 12x - 30 = 16 → 14x - 30 = 16 Add 30 to both sides: 14x = 46 Divide by 14: x = 46/14 = 23/7 = 3.5 Substitute x = 3.5 into y = 4x - 10: y =…
   Full step-by-step solution

   Step 1: Solve the second equation for y: 4x - y = 10 → -y = 10 - 4x → y = 4x - 10 Step 2: Substitute y = 4x - 10 into the first equation: 2x + 3(4x - 10) = 16 Step 3: Simplify: 2x + 12x - 30 = 16 → 14x - 30 = 16 Step 4: Add 30 to both sides: 14x = 46 Step 5: Divide by 14: x = 46/14 = 23/7 = 3.5 Step 6: Substitute x = 3.5 into y = 4x - 10: y = 4(3.5) - 10 = 14 - 10 = 4 Step 7: Check: 2(3.5) + 3(4) = 7 + 12 = 19 (slight rounding error) Let's recalculate more precisely: x = 23/7 ≈ 3.2857, y = 4(23/7) - 10 = 92/7 - 70/7 = 22/7 ≈ 3.1429 Check: 2(23/7) + 3(22/7) = 46/7 + 66/7 = 112/7 = 16 ✓ 4(23/7) - 22/7 = 92/7 - 22/7 = 70/7 = 10 ✓ The exact solution is x = 23/7, y = 22/7

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (0,8). A circle is drawn such that its diameter is the hypotenuse of the triangle. What is the area of the circle? (Use π = 3.14) Answer: 78.5 Solution: Find the length of the hypotenuse using the Pythagorean theorem. The legs of the triangle are 6 units (from (0,0) to (6,0)) and 8 units (from (0,0) to (0,8)).
   Full step-by-step solution

   Step 1: Find the length of the hypotenuse using the Pythagorean theorem. The legs of the triangle are 6 units (from (0,0) to (6,0)) and 8 units (from (0,0) to (0,8)). Hypotenuse = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 units. Step 2: The hypotenuse is the diameter of the circle. Diameter = 10 units Radius = diameter/2 = 10/2 = 5 units Step 3: Calculate the area of the circle. Area = π × radius^2 = 3.14 × 5^2 = 3.14 × 25 = 78.5 square units The answer is 78.5.

5. Emma is helping her school's debate team raise money by selling two types of gift baskets. A small basket contains 3 chocolates and costs $7, while a large basket contains 5 chocolates and costs $11. The team sold a total of 31 baskets and collected $265. How many small baskets and how many large baskets did they sell? Answer: small baskets = 19, large baskets = 12 Solution: Let s = number of small baskets and l = number of large baskets.
   Full step-by-step solution

   Let s = number of small baskets and l = number of large baskets. Equation 1 (total baskets): s + l = 31 Equation 2 (total money): 7s + 11l = 265 Solve the first equation for s: s = 31 - l Substitute into the second equation: 7(31 - l) + 11l = 265 Simplify: 217 - 7l + 11l = 265 Combine like terms: 217 + 4l = 265 Subtract 217 from both sides: 4l = 48 Divide by 4: l = 12 Substitute back to find s: s = 31 - 12 = 19 They sold 19 small baskets and 12 large baskets.

6. Noah and Ava are selling tickets for a school carnival. Adult tickets cost $6 each and child tickets cost $4 each. They sold a total of 116 tickets and collected $576. How many adult tickets and how many child tickets did they sell? Answer: Adult tickets: 56, Child tickets: 60 Solution: Let a = number of adult tickets, c = number of child tickets. Write the equations. Total tickets: a + c = 116 Total money: 6a + 4c = 576 Solve by elimination.
   Full step-by-step solution

   Step 1: Let a = number of adult tickets, c = number of child tickets. Step 2: Write the equations. Total tickets: a + c = 116 Total money: 6a + 4c = 576 Step 3: Solve by elimination. Multiply the first equation by 4: 4a + 4c = 464. Step 4: Subtract this from the second equation: (6a + 4c) - (4a + 4c) = 576 - 464, so 2a = 112, a = 56. Step 5: Substitute a = 56 into a + c = 116: 56 + c = 116, so c = 60. Step 6: Check: 56 + 60 = 116 tickets, and 6(56) + 4(60) = 336 + 240 = 576 dollars. Correct. The answer is 56 adult tickets and 60 child tickets.

7. A space probe is traveling toward Mars at a constant speed. The probe is currently 3.6 × 10^7 kilometers from Earth and moving away at 2.4 × 10^4 kilometers per hour. At the same time, a supply ship is launched from Earth to catch up with the probe. The supply ship travels at 4.8 × 10^4 kilometers per hour. How many hours after the supply ship launches will it catch up to the space probe? Answer: 1500 Solution: Let h represent the number of hours after the supply ship launches when they meet. The probe's head start is 3.6 × 10^7 km.
   Full step-by-step solution

   Step 1: Let h represent the number of hours after the supply ship launches when they meet. Step 2: The probe's head start is 3.6 × 10^7 km. Step 3: During h hours, the probe travels: (2.4 × 10^4) × h km Step 4: During h hours, the supply ship travels: (4.8 × 10^4) × h km Step 5: When they meet: supply ship distance = probe head start + probe distance traveled Step 6: (4.8 × 10^4)h = 3.6 × 10^7 + (2.4 × 10^4)h Step 7: Subtract (2.4 × 10^4)h from both sides: (2.4 × 10^4)h = 3.6 × 10^7 Step 8: Divide both sides by 2.4 × 10^4: h = (3.6 × 10^7) ÷ (2.4 × 10^4) Step 9: h = (3.6 ÷ 2.4) × (10^7 ÷ 10^4) Step 10: h = 1.5 × 10^3 Step 11: h = 1500 The answer is 1500 hours.

8. A rectangular garden is drawn on a coordinate plane with corners at points (1, 2), (7, 2), (7, 6), and (1, 6). A diagonal path is drawn from the bottom-left corner to the top-right corner, and a second diagonal path is drawn from the bottom-right corner to the top-left corner. These diagonals intersect at the center of the rectangle, creating four smaller triangles. What is the area of one of these four triangles? Answer: 6 Solution: Find the dimensions of the rectangle. The bottom-left corner is at (1, 2) and the bottom-right corner is at (7, 2). The length is 7 - 1 = 6 units.
   Full step-by-step solution

   Step 1: Find the dimensions of the rectangle. The bottom-left corner is at (1, 2) and the bottom-right corner is at (7, 2). The length is 7 - 1 = 6 units. The bottom-left corner is at (1, 2) and the top-left corner is at (1, 6). The width is 6 - 2 = 4 units. Step 2: Calculate the area of the entire rectangle. Area = length × width = 6 × 4 = 24 square units. Step 3: Understand how the diagonals divide the rectangle. The two diagonals intersect at the center and divide the rectangle into four smaller triangles of equal area. Step 4: Calculate the area of one small triangle. Total area = 24 square units Number of equal triangles = 4 Area of one triangle = 24 ÷ 4 = 6 square units. The answer is 6.