Systems Word Problems

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Answer Key & Explanations

Systems Word Problems · Grade 8 · Worksheet 3

1. 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.

2. Charlotte and Isabella are selling cookies for a fundraiser. Charlotte sells boxes of chocolate chip cookies for $9 each, and Isabella sells boxes of oatmeal cookies for $11 each. Together, they sold 48 boxes and earned a total of $468. How many boxes did each person sell? Answer: Charlotte sold 30 boxes, Isabella sold 18 boxes Solution: Let c = number of boxes Charlotte sold, and i = number of boxes Isabella sold. Write the equations. Total boxes: c + i = 48 Total money: 9c + 11i = 468 Solve by elimination.
   Full step-by-step solution

   Step 1: Let c = number of boxes Charlotte sold, and i = number of boxes Isabella sold. Step 2: Write the equations. Total boxes: c + i = 48 Total money: 9c + 11i = 468 Step 3: Solve by elimination. Multiply the first equation by 9: 9c + 9i = 432. Step 4: Subtract from the second equation: (9c + 11i) - (9c + 9i) = 468 - 432, so 2i = 36, i = 18. Step 5: Substitute i = 18 into c + i = 48: c + 18 = 48, so c = 30. Step 6: Check: 30 + 18 = 48 boxes, and 9(30) + 11(18) = 270 + 198 = 468 dollars. Correct. The answer is Charlotte sold 30 boxes and Isabella sold 18 boxes.

3. Noah and Ava are collecting donations for a charity. Noah collected 16 more donations than Ava. Together, they collected 126 donations. How many donations did each person collect? Answer: Noah: 71, Ava: 55 Solution: Let a = number of donations Ava collected. Then Noah collected a + 16 donations. Together: a + (a + 16) = 126.
   Full step-by-step solution

   Let a = number of donations Ava collected. Then Noah collected a + 16 donations. Together: a + (a + 16) = 126. Simplify: 2a + 16 = 126. Subtract 16: 2a = 110. Divide by 2: a = 55. So Ava collected 55 donations. Noah collected 55 + 16 = 71 donations. Check: 55 + 71 = 126. Answer: Noah collected 71, Ava collected 55.

4. 2x + 3y = 16 and 4x - y = 18, find x and y Answer: x = 5, y = 2 Solution: Step 1: Write the system: 2x + 3y = 16 and 4x - y = 18 Step 2: Multiply the second equation by 3 to eliminate y: 3(4x - y) = 3(18) → 12x - 3y = 54 Step 3: Add this to the first equation: (2x + 3y) + (12x - 3y) = 16 + 54 → 14x = 70 Step 4: Solve for x: x = 70 ÷ 14 = 5 Step 5: Substitute x = 5…
   Full step-by-step solution

   Step 1: Write the system: 2x + 3y = 16 and 4x - y = 18 Step 2: Multiply the second equation by 3 to eliminate y: 3(4x - y) = 3(18) → 12x - 3y = 54 Step 3: Add this to the first equation: (2x + 3y) + (12x - 3y) = 16 + 54 → 14x = 70 Step 4: Solve for x: x = 70 ÷ 14 = 5 Step 5: Substitute x = 5 into the first equation: 2(5) + 3y = 16 → 10 + 3y = 16 Step 6: Solve for y: 3y = 6 → y = 2 Step 7: Verify with second equation: 4(5) - 2 = 20 - 2 = 18 ✓ The solution is x = 5, y = 2.

5. Matiu is designing a rectangular garden on a coordinate grid. The garden has corners at points A(2, 2), B(2, 10), C(14, 10), and D(14, 2). He wants to install a diagonal water pipe from corner A to corner C, and another diagonal pipe from corner B to corner D. The two pipes cross at the center of the rectangle, dividing the garden into four triangular sections. Hana plans to plant different flowers in each triangular section. What is the area, in square units, of one of the four triangular sections? Answer: 24 Solution: Find the length and width of the rectangle. The bottom side goes from A(2,2) to D(14,2), so length = 14 - 2 = 12 units. The left side goes from A(2,2) to B(2,10), so width = 10 - 2 = 8 units.
   Full step-by-step solution

   Step 1: Find the length and width of the rectangle. The bottom side goes from A(2,2) to D(14,2), so length = 14 - 2 = 12 units. The left side goes from A(2,2) to B(2,10), so width = 10 - 2 = 8 units. Step 2: Calculate the area of the entire rectangle. Area = length x width = 12 x 8 = 96 square units. Step 3: Understand how the diagonals divide the rectangle. The two diagonals of a rectangle intersect at the center and are equal in length. They divide the rectangle into four triangles that are all equal in area. Step 4: Find the area of one triangle. Since the four triangles have equal area, area of one triangle = total area / 4 = 96 / 4 = 24 square units. The answer is 24.

6. 2x + 3y = 16 and 5x - 2y = 21, find x and y Answer: x = 5, y = 2 Solution: Multiply the first equation by 2: 2(2x + 3y) = 2(16) → 4x + 6y = 32 Multiply the second equation by 3: 3(5x - 2y) = 3(21) → 15x - 6y = 63 Add the two new equations: (4x + 6y) + (15x - 6y) = 32 + 63 → 19x = 95 Solve for x: x = 95 ÷ 19 = 5 Substitute x = 5 into the first equation: 2(5) + 3y = 16 →…
   Full step-by-step solution

   Step 1: Multiply the first equation by 2: 2(2x + 3y) = 2(16) → 4x + 6y = 32 Step 2: Multiply the second equation by 3: 3(5x - 2y) = 3(21) → 15x - 6y = 63 Step 3: Add the two new equations: (4x + 6y) + (15x - 6y) = 32 + 63 → 19x = 95 Step 4: Solve for x: x = 95 ÷ 19 = 5 Step 5: Substitute x = 5 into the first equation: 2(5) + 3y = 16 → 10 + 3y = 16 Step 6: Solve for y: 3y = 6 → y = 2 Step 7: Check in second equation: 5(5) - 2(2) = 25 - 4 = 21 ✓ The solution is x = 5, y = 2.

7. Isabella is helping her school's drama club prepare for a play. They need to buy tickets for two types of seats: orchestra seats cost $14 each and balcony seats cost $9 each. The total number of tickets sold was 200, and the total amount of money collected was $2,430. How many orchestra tickets and how many balcony tickets were sold? Answer: 126 orchestra tickets and 74 balcony tickets Solution: Let x = number of orchestra tickets and y = number of balcony tickets.
   Full step-by-step solution

   Let x = number of orchestra tickets and y = number of balcony tickets. Step 1: Write the equation for total tickets: x + y = 200 Step 2: Write the equation for total money: 14x + 9y = 2430 Step 3: Solve the first equation for y: y = 200 - x Step 4: Substitute into the second equation: 14x + 9(200 - x) = 2430 Step 5: Distribute and simplify: 14x + 1800 - 9x = 2430 5x + 1800 = 2430 Step 6: Subtract 1800 from both sides: 5x = 630 Step 7: Divide by 5: x = 126 Step 8: Substitute back to find y: y = 200 - 126 = 74 There were 126 orchestra tickets and 74 balcony tickets sold.