Systems Word Problems

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Systems Word Problems · Grade 8 · Worksheet 1

1. 2x + 3y = 19 and 4x - 2y = 18, find x and y Answer: x = 5, y = 3 Solution: Step 1: Multiply the first equation by 2 to match the x-coefficient with the second equation: 2(2x + 3y) = 2(19) → 4x + 6y = 38 Step 2: Subtract the second equation from this result: (4x + 6y) - (4x - 2y) = 38 - 18 → 4x + 6y - 4x + 2y = 20 → 8y = 20 Step 3: Solve for y: y = 20 ÷ 8 = 2.5 Step 4:…
   Full step-by-step solution

   Step 1: Multiply the first equation by 2 to match the x-coefficient with the second equation: 2(2x + 3y) = 2(19) → 4x + 6y = 38 Step 2: Subtract the second equation from this result: (4x + 6y) - (4x - 2y) = 38 - 18 → 4x + 6y - 4x + 2y = 20 → 8y = 20 Step 3: Solve for y: y = 20 ÷ 8 = 2.5 Step 4: Substitute y = 2.5 into the first equation: 2x + 3(2.5) = 19 → 2x + 7.5 = 19 → 2x = 11.5 → x = 5.75 Step 5: Verify with second equation: 4(5.75) - 2(2.5) = 23 - 5 = 18 ✓ The solution is x = 5.75, y = 2.5

2. Emma is planning a school science fair and needs to buy materials for two different experiments. For the first experiment, she needs to buy test tubes that cost $4 each. For the second experiment, she needs to buy beakers that cost $6 each. Her total budget for these materials is $180, and she needs a total of 35 items. How many test tubes and how many beakers should Emma buy to spend exactly her budget and get exactly 35 items? Answer: 15 Solution: Let x be the number of test tubes and y be the number of beakers. Write the equation for the total number of items: x + y = 35 Write the equation for the total cost: 4x + 6y = 180 Solve the first equation for x: x = 35 - y Substitute into the second equation: 4(35 - y) + 6y = 180 Simplify: 140 -…
   Full step-by-step solution

   Let x be the number of test tubes and y be the number of beakers. Step 1: Write the equation for the total number of items: x + y = 35 Step 2: Write the equation for the total cost: 4x + 6y = 180 Step 3: Solve the first equation for x: x = 35 - y Step 4: Substitute into the second equation: 4(35 - y) + 6y = 180 Step 5: Simplify: 140 - 4y + 6y = 180 Step 6: Combine like terms: 140 + 2y = 180 Step 7: Subtract 140 from both sides: 2y = 40 Step 8: Divide by 2: y = 20 Step 9: Substitute back to find x: x = 35 - 20 = 15 Emma should buy 15 test tubes and 20 beakers. The question asks for test tubes, so the answer is 15.

3. 2x + 3y = 12 and 4x - y = 10, find x and y Answer: x = 3, y = 2 Solution: (1) 2x + 3y = 12 (2) 4x - y = 10 Solve one equation for one variable. From equation (2): 4x - y = 10 Add y to both sides: 4x = 10 + y Subtract 10 from both sides: y = 4x - 10 Substitute y = 4x - 10 into equation (1).
   Full step-by-step solution

   We are given the system of equations: (1) 2x + 3y = 12 (2) 4x - y = 10 Step 1: Solve one equation for one variable. From equation (2): 4x - y = 10 Add y to both sides: 4x = 10 + y Subtract 10 from both sides: y = 4x - 10 Step 2: Substitute y = 4x - 10 into equation (1). Equation (1): 2x + 3y = 12 Replace y with (4x - 10): 2x + 3(4x - 10) = 12 Step 3: Simplify and solve for x. 2x + 12x - 30 = 12 14x - 30 = 12 Add 30 to both sides: 14x = 42 Divide both sides by 14: x = 42/14 x = 3 Step 4: Substitute x = 3 into y = 4x - 10. y = 4(3) - 10 y = 12 - 10 y = 2 Step 5: Check the solution in both original equations. In (1): 2(3) + 3(2) = 6 + 6 = 12 ✓ In (2): 4(3) - 2 = 12 - 2 = 10 ✓ Final answer: x = 3, y = 2

4. Maya is planning a school trip to the science museum. The museum charges $12 per student and $18 per adult. The total cost for the trip is $600, and there are 40 people going in total. How many students and how many adults are going on the trip? Answer: 20 Solution: Let s be the number of students and a be the number of adults. Write the equation for the total number of people: s + a = 40 Write the equation for the total cost: 12s + 18a = 600 Solve the first equation for s: s = 40 - a Substitute this into the second equation: 12(40 - a) + 18a = 600…
   Full step-by-step solution

   Step 1: Let s be the number of students and a be the number of adults. Step 2: Write the equation for the total number of people: s + a = 40 Step 3: Write the equation for the total cost: 12s + 18a = 600 Step 4: Solve the first equation for s: s = 40 - a Step 5: Substitute this into the second equation: 12(40 - a) + 18a = 600 Step 6: Simplify: 480 - 12a + 18a = 600 Step 7: Combine like terms: 480 + 6a = 600 Step 8: Subtract 480 from both sides: 6a = 120 Step 9: Divide both sides by 6: a = 20 Step 10: Substitute back to find s: s = 40 - 20 = 20 There are 20 students and 20 adults going on the trip.

5. Maya is planning a community garden and needs to determine how many tomato plants and pepper plants to buy. Tomato plants cost $4 each and pepper plants cost $3 each. Her total budget for plants is $150. Additionally, she knows that the number of tomato plants should be twice the number of pepper plants. How many tomato plants and how many pepper plants should Maya buy? Answer: x = 30, y = 15 Solution: The key is to identify the two conditions: one that relates the quantities of the two items, and another that relates to the total cost.
   Full step-by-step solution

   In real-world situations involving two variables with a specific relationship and a budget constraint, we can use systems of equations. The key is to identify the two conditions: one that relates the quantities of the two items, and another that relates to the total cost. By substituting one equation into the other, we can solve for both unknown quantities.

6. 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.

7. Aroha and Tane are selling two types of raffle tickets. Premium tickets cost $12 each and Standard tickets cost $7 each. They sell 95 tickets in total and collect $890. How many Premium tickets and how many Standard tickets did they sell? Answer: Premium: 45, Standard: 50 Solution: Let p = number of Premium tickets, s = number of Standard tickets. Total tickets equation: p + s = 95. Total money equation: 12p + 7s = 890.
   Full step-by-step solution

   Step 1: Let p = number of Premium tickets, s = number of Standard tickets. Step 2: Total tickets equation: p + s = 95. Step 3: Total money equation: 12p + 7s = 890. Step 4: Solve by elimination. Multiply the first equation by 7: 7p + 7s = 665. Step 5: Subtract from the second equation: (12p + 7s) - (7p + 7s) = 890 - 665, so 5p = 225, p = 45. Step 6: Substitute p = 45 into p + s = 95: 45 + s = 95, so s = 50. Step 7: Check: 45 + 50 = 95 tickets, and 12(45) + 7(50) = 540 + 350 = 890 dollars. Correct. The answer is Premium: 45, Standard: 50.