Systems by Substitution

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Systems by Substitution · Grade 8 · Worksheet 3

1. Charlotte is helping her school organize a book fair. She needs to order fiction and nonfiction books. Fiction books cost $9 each and nonfiction books cost $14 each. The school has a budget of exactly $330 for books. Charlotte also knows that the total number of books ordered must be exactly 30. How many fiction books and how many nonfiction books should Charlotte order to spend the entire budget? Answer: 18 fiction books and 12 nonfiction books Solution: Let f = number of fiction books, n = number of nonfiction books.
   Full step-by-step solution

   Let f = number of fiction books, n = number of nonfiction books. Equation 1 (total books): f + n = 30 Equation 2 (total cost): 9f + 14n = 330 Solve equation 1 for f: f = 30 - n Substitute into equation 2: 9(30 - n) + 14n = 330 Simplify: 270 - 9n + 14n = 330 Combine like terms: 270 + 5n = 330 Subtract 270 from both sides: 5n = 60 Divide by 5: n = 12 Substitute n = 12 into f = 30 - n: f = 30 - 12 = 18 Check: 9(18) + 14(12) = 162 + 168 = 330 (correct) The answer is 18 fiction books and 12 nonfiction books.

2. 3x - 2y = 16; y = 2x - 10; x = ? Answer: 4 Solution: Start with the system: 3x - 2y = 16 and y = 2x - 10 Substitute y = 2x - 10 into the first equation: 3x - 2(2x - 10) = 16 Distribute the -2: 3x - 4x + 20 = 16 Combine like terms: -x + 20 = 16 Subtract 20 from both sides: -x = -4 Multiply both sides by -1: x = 4 The answer is 4.
   Full step-by-step solution

   Step 1: Start with the system: 3x - 2y = 16 and y = 2x - 10 Step 2: Substitute y = 2x - 10 into the first equation: 3x - 2(2x - 10) = 16 Step 3: Distribute the -2: 3x - 4x + 20 = 16 Step 4: Combine like terms: -x + 20 = 16 Step 5: Subtract 20 from both sides: -x = -4 Step 6: Multiply both sides by -1: x = 4 The answer is 4.

3. Matiu is helping to organize the school's sports equipment. He needs to sort basketballs and soccer balls. Each basketball weighs 2 kilograms, and each soccer ball weighs 1 kilogram. The total weight of all the balls is 74 kilograms. If there are 48 balls in total, how many basketballs and how many soccer balls are there? Answer: 26 basketballs and 22 soccer balls Solution: Let b = number of basketballs and s = number of soccer balls.
   Full step-by-step solution

   Let b = number of basketballs and s = number of soccer balls. Equation 1 (total balls): b + s = 48 Equation 2 (total weight): 2b + 1s = 74 Solve Equation 1 for s: s = 48 - b Substitute into Equation 2: 2b + (48 - b) = 74 Simplify: 2b + 48 - b = 74 Combine like terms: b + 48 = 74 Subtract 48 from both sides: b = 26 Now find s: s = 48 - 26 = 22 So there are 26 basketballs and 22 soccer balls.

4. Mere is making gift bags for a school event. She has two types of bags: small bags that hold 2 treats each and large bags that hold 4 treats each. She needs to pack a total of 48 treats using exactly 16 bags. How many small bags and how many large bags should Mere use? Answer: 8 small bags and 8 large bags Solution: Let s = number of small bags, and l = number of large bags.
   Full step-by-step solution

   Let s = number of small bags, and l = number of large bags. Equation 1 (total bags): s + l = 16 Equation 2 (total treats): 2s + 4l = 48 Solve equation 1 for s: s = 16 - l Substitute into equation 2: 2(16 - l) + 4l = 48 Simplify: 32 - 2l + 4l = 48 Combine like terms: 32 + 2l = 48 Subtract 32 from both sides: 2l = 16 Divide by 2: l = 8 Now substitute l = 8 into s = 16 - l: s = 16 - 8 = 8 So Mere needs 8 small bags and 8 large bags. Check: 8 + 8 = 16 bags total, and 2(8) + 4(8) = 16 + 32 = 48 treats. The answer is 8 small bags and 8 large bags.

5. 2x + 7y = 32; y = 3x - 2; x = ? Answer: 2 Solution: Start with the system: 2x + 7y = 32 and y = 3x - 2. Substitute y = 3x - 2 into the first equation: 2x + 7(3x - 2) = 32. Distribute the 7: 2x + 21x - 14 = 32.
   Full step-by-step solution

   Step 1: Start with the system: 2x + 7y = 32 and y = 3x - 2. Step 2: Substitute y = 3x - 2 into the first equation: 2x + 7(3x - 2) = 32. Step 3: Distribute the 7: 2x + 21x - 14 = 32. Step 4: Combine like terms: 23x - 14 = 32. Step 5: Add 14 to both sides: 23x = 46. Step 6: Divide both sides by 23: x = 2. The value of x is 2.

6. y = 2x - 7; 5x + 2y = 22; x = ? Answer: 4 Solution: Start with the system: y = 2x - 7 and 5x + 2y = 22. Substitute y = 2x - 7 into the second equation: 5x + 2(2x - 7) = 22. Distribute the 2: 5x + 4x - 14 = 22.
   Full step-by-step solution

   Step 1: Start with the system: y = 2x - 7 and 5x + 2y = 22. Step 2: Substitute y = 2x - 7 into the second equation: 5x + 2(2x - 7) = 22. Step 3: Distribute the 2: 5x + 4x - 14 = 22. Step 4: Combine like terms: 9x - 14 = 22. Step 5: Add 14 to both sides: 9x = 36. Step 6: Divide both sides by 9: x = 4. The value of x is 4.

7. Liam is organizing 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 amount 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 equations from the problem. s + a = 120 ... (1) 3s + 5a = 500 ...
   Full step-by-step solution

   Let’s define variables: Let s = number of student tickets Let a = number of adult tickets --- **Step 1: Write the equations from the problem.** From the total number of tickets: s + a = 120 ... (1) From the total money collected: 3s + 5a = 500 ... (2) --- **Step 2: Solve for one variable using the first equation.** From (1): s = 120 - a --- **Step 3: Substitute into the second equation.** Substitute s into (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 + 70 × 5 = 150 + 350 = 500 ✔ --- **Final answer:** 50 student tickets, 70 adult tickets

8. 3x - 2y = 8; y = 2x - 5; x = ? Answer: 2 Solution: Start with the equations: 3x - 2y = 8 and y = 2x - 5. Substitute (2x - 5) for y in the first equation: 3x - 2(2x - 5) = 8. Distribute the -2: 3x - 4x + 10 = 8.
   Full step-by-step solution

   Step 1: Start with the equations: 3x - 2y = 8 and y = 2x - 5. Step 2: Substitute (2x - 5) for y in the first equation: 3x - 2(2x - 5) = 8. Step 3: Distribute the -2: 3x - 4x + 10 = 8. Step 4: Combine like terms: -x + 10 = 8. Step 5: Subtract 10 from both sides: -x = -2. Step 6: Multiply both sides by -1: x = 2. The value of x is 2.