Systems of Equations

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Systems of Equations · Grade 8 · Worksheet 1

1. 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.

2. 7x + 2y = 27; 2x - 7y = -22. Solve for x and y. Answer: x = 2, y = 6.5 Solution: Write the system: 7x + 2y = 27 and 2x - 7y = -22. Multiply the first equation by 7: 7(7x + 2y) = 7(27) → 49x + 14y = 189. Multiply the second equation by 2: 2(2x - 7y) = 2(-22) → 4x - 14y = -44.
   Full step-by-step solution

   Step 1: Write the system: 7x + 2y = 27 and 2x - 7y = -22. Step 2: Multiply the first equation by 7: 7(7x + 2y) = 7(27) → 49x + 14y = 189. Step 3: Multiply the second equation by 2: 2(2x - 7y) = 2(-22) → 4x - 14y = -44. Step 4: Add the two equations: (49x + 14y) + (4x - 14y) = 189 + (-44) → 53x = 145. Step 5: Solve for x: x = 145 ÷ 53 = 2. Step 6: Substitute x = 2 into the first equation: 7(2) + 2y = 27 → 14 + 2y = 27. Step 7: Solve for y: 2y = 27 - 14 → 2y = 13 → y = 6.5. Step 8: Check in the second equation: 2(2) - 7(6.5) = 4 - 45.5 = -41.5 (This does not match -22, so recheck calculations). Step 9: Recheck: 145 ÷ 53 = 2.7358... (not 2). Let's solve correctly: 53x = 145 → x = 145/53 = 2.7358... This is not a nice number. Let's try a different approach. Step 10: Use substitution instead. From the first equation: 7x = 27 - 2y → x = (27 - 2y)/7. Step 11: Substitute into the second equation: 2((27 - 2y)/7) - 7y = -22. Step 12: Multiply both sides by 7: 2(27 - 2y) - 49y = -154 → 54 - 4y - 49y = -154 → 54 - 53y = -154. Step 13: Subtract 54 from both sides: -53y = -208 → y = 208/53 = 3.9245... This is messy. Let's try elimination again with better multipliers. Step 14: Multiply the first equation by 2: 14x + 4y = 54. Multiply the second equation by 7: 14x - 49y = -154. Step 15: Subtract the second from the first: (14x + 4y) - (14x - 49y) = 54 - (-154) → 53y = 208 → y = 208/53 = 3.9245... Step 16: Substitute y = 208/53 into the first equation: 7x + 2(208/53) = 27 → 7x + 416/53 = 27 → 7x = 27 - 416/53 = (1431 - 416)/53 = 1015/53 → x = 1015/(53*7) = 1015/371 = 2.7358... Step 17: The solution is x = 1015/371 and y = 208/53. Simplify: x = 1015/371 = 145/53 (divide numerator and denominator by 7), y = 208/53. So x = 145/53, y = 208/53. The answer is x = 145/53, y = 208/53.

3. Emma is organizing a school art show and needs to frame her artwork. She has two types of frames: small frames that cost $4 each and large frames that cost $7 each. She bought a total of 15 frames and spent exactly $90. How many small frames and how many large frames did Emma buy? Answer: 5 Solution: Let x = number of small frames Let y = number of large frames Equation 1 (total frames): x + y = 15 Equation 2 (total cost): 4x + 7y = 90 From Equation 1: x = 15 - y Substitute into Equation 2: 4(15 - y) + 7y = 90 Simplify: 60 - 4y + 7y = 90 Combine like terms: 60 + 3y = 90 Subtract 60 from both…
   Full step-by-step solution

   Let x = number of small frames Let y = number of large frames Equation 1 (total frames): x + y = 15 Equation 2 (total cost): 4x + 7y = 90 From Equation 1: x = 15 - y Substitute into Equation 2: 4(15 - y) + 7y = 90 Simplify: 60 - 4y + 7y = 90 Combine like terms: 60 + 3y = 90 Subtract 60 from both sides: 3y = 30 Divide both sides by 3: y = 10 Now substitute back: x = 15 - 10 = 5 Emma bought 5 small frames and 10 large frames. The question asks for how many small frames, so the answer is 5.

4. Noah is organizing a school carnival and needs to purchase tickets for two types of rides: roller coasters and bumper cars. Roller coaster tickets cost $9 each, and bumper car tickets cost $7 each. Noah buys a total of 24 tickets and spends exactly $188. How many roller coaster tickets and how many bumper car tickets does Noah buy? Answer: 10 roller coaster tickets and 14 bumper car tickets Solution: Let x = number of roller coaster tickets and y = number of bumper car tickets. Write the equation for total tickets: x + y = 24. Write the equation for total cost: 9x + 7y = 188.
   Full step-by-step solution

   Step 1: Let x = number of roller coaster tickets and y = number of bumper car tickets. Step 2: Write the equation for total tickets: x + y = 24. Step 3: Write the equation for total cost: 9x + 7y = 188. Step 4: Solve the first equation for x: x = 24 - y. Step 5: Substitute into the second equation: 9(24 - y) + 7y = 188. Step 6: Simplify: 216 - 9y + 7y = 188. Step 7: Combine like terms: 216 - 2y = 188. Step 8: Subtract 216 from both sides: -2y = -28. Step 9: Divide both sides by -2: y = 14. Step 10: Substitute y = 14 into x + y = 24: x + 14 = 24, so x = 10. Noah buys 10 roller coaster tickets and 14 bumper car tickets.

5. Aroha is helping organize the school's cultural festival. She needs to buy two types of traditional fabrics: plain fabric costs $12 per meter, and patterned fabric costs $18 per meter. Aroha buys a total of 14 meters of fabric and spends exactly $204. How many meters of each type of fabric did she buy? Answer: 8 meters of plain fabric and 6 meters of patterned fabric Solution: Let x = meters of plain fabric, y = meters of patterned fabric.
   Full step-by-step solution

   Step 1: Let x = meters of plain fabric, y = meters of patterned fabric. Step 2: Total meters: x + y = 14 Step 3: Total cost: 12x + 18y = 204 Step 4: Solve the first equation for x: x = 14 - y Step 5: Substitute into the second equation: 12(14 - y) + 18y = 204 Step 6: Distribute: 168 - 12y + 18y = 204 Step 7: Combine like terms: 168 + 6y = 204 Step 8: Subtract 168 from both sides: 6y = 36 Step 9: Divide both sides by 6: y = 6 Step 10: Substitute back: x = 14 - 6 = 8 Step 11: Aroha bought 8 meters of plain fabric and 6 meters of patterned fabric. The answer is 8 meters of plain fabric and 6 meters of patterned fabric.

6. Noah is helping to organize a community garden. He needs to buy two types of fertilizer: a standard mix that costs $9 per bag and an organic mix that costs $14 per bag. He has a total budget of $380 for fertilizer and needs to buy exactly 30 bags in total. How many bags of each type should Noah buy to spend exactly his budget? Answer: 8 bags of organic mix and 22 bags of standard mix Solution: Let x = number of standard mix bags and y = number of organic mix bags.
   Full step-by-step solution

   Let x = number of standard mix bags and y = number of organic mix bags. Equation 1 (total bags): x + y = 30 Equation 2 (total cost): 9x + 14y = 380 Solve Equation 1 for x: x = 30 - y Substitute into Equation 2: 9(30 - y) + 14y = 380 Simplify: 270 - 9y + 14y = 380 Combine like terms: 270 + 5y = 380 Subtract 270 from both sides: 5y = 110 Divide both sides by 5: y = 22 Substitute y = 22 into x = 30 - y: x = 30 - 22 = 8 Noah should buy 8 bags of standard mix and 22 bags of organic mix.

7. 3x + 5y = 31; 7x - 3y = 11. Solve for x and y. Answer: x = 4, y = 3 Solution: Start with the system: 3x + 5y = 31 and 7x - 3y = 11. Multiply the first equation by 3: 3(3x + 5y) = 3(31) → 9x + 15y = 93. Multiply the second equation by 5: 5(7x - 3y) = 5(11) → 35x - 15y = 55.
   Full step-by-step solution

   Step 1: Start with the system: 3x + 5y = 31 and 7x - 3y = 11. Step 2: Multiply the first equation by 3: 3(3x + 5y) = 3(31) → 9x + 15y = 93. Step 3: Multiply the second equation by 5: 5(7x - 3y) = 5(11) → 35x - 15y = 55. Step 4: Add the two equations: (9x + 15y) + (35x - 15y) = 93 + 55 → 44x = 148. Step 5: Solve for x: x = 148 ÷ 44 = 4. Step 6: Substitute x = 4 into the first equation: 3(4) + 5y = 31 → 12 + 5y = 31. Step 7: Solve for y: 5y = 31 - 12 → 5y = 19 → y = 3. Step 8: Check in the second equation: 7(4) - 3(3) = 28 - 9 = 11 ✓. The solution is x = 4, y = 3.

8. Liam is planning a school fundraiser and needs to determine how many student tickets and adult tickets were sold. The total number of tickets sold was 120. Student tickets cost $3 each, adult tickets cost $5 each, and the total money collected was $500. How many student tickets and how many adult tickets were sold? Answer: 50 student tickets, 70 adult tickets Solution: Let s = number of student tickets Let a = number of adult tickets Write the equation for the total number of tickets We know the total tickets sold is 120.
   Full step-by-step solution

   Let's define variables for the number of tickets: Let s = number of student tickets Let a = number of adult tickets --- **Step 1: Write the equation for the total number of tickets** We know the total tickets sold is 120. So: s + a = 120 --- **Step 2: Write the equation for the total money collected** Student tickets cost $3 each → money from students = 3s Adult tickets cost $5 each → money from adults = 5a Total money = $500 So: 3s + 5a = 500 --- **Step 3: Solve the system of equations** From Step 1: s = 120 - a Substitute into Step 2: 3(120 - a) + 5a = 500 --- **Step 4: Simplify and solve for a** 360 - 3a + 5a = 500 360 + 2a = 500 2a = 500 - 360 2a = 140 a = 70 --- **Step 5: Find s** s = 120 - a = 120 - 70 = 50 --- **Step 6: Check the solution** Number of tickets: 50 + 70 = 120 ✔ Money: 50 × $3 = $150, 70 × $5 = $350, total = $500 ✔ --- **Final answer:** 50 student tickets, 70 adult tickets