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Linear Systems 3x3

Grade 12 · Algebra · Worksheet 2

  1. 3x + y - 5z = 7, x - 3y + z = -5, 5x - y + 3z = 11 Answer: ______________
  2. x + y + z = 8, 2x - y + z = 7, x - 2y + 2z = 4 Answer: ______________
  3. Liam is tracking his weekly earnings from three different jobs: tutoring, dog walking, and yard work. This week, he earned 3 times as much from tutoring as from dog walking, and his yard work earnings were $7 more than his dog walking earnings. The total amount he earned from all three jobs was $47. How much did Liam earn from dog walking? Answer: ______________
  4. Solve: 3x + 5y - z = 7, x - 3y + 5z = -1, 5x + y - 3z = 11 Answer: ______________
  5. Aroha is managing a small business that produces three types of candles: lavender, sandalwood, and vanilla. In one production run, she uses wax, fragrance oil, and wicks. For lavender candles, she uses 3 units of wax, 2 units of fragrance oil, and 1 unit of wicks. For sandalwood candles, she uses 4 units of wax, 1 unit of fragrance oil, and 2 units of wicks. For vanilla candles, she uses 2 units of wax, 3 units of fragrance oil, and 1 unit of wicks. If Aroha used 41 total units of wax, 29 total units of fragrance oil, and 18 total units of wicks, how many vanilla candles did she produce? Answer: ______________
  6. 2x + 3y - z = 7, x - y + 2z = 1, 3x + y + z = 9 Answer: ______________
  7. Solve: 3x + 2y - 4z = 7, 2x - 3y + z = -12, x + 4y - 2z = 17 Answer: ______________
  8. Solve: 2x + y - z = 6, x - 3y + 2z = -1, 3x + 2y + z = 11 Answer: ______________
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Answer Key & Explanations

Linear Systems 3x3 · Grade 12 · Worksheet 2

  1. 3x + y - 5z = 7, x - 3y + z = -5, 5x - y + 3z = 11 Answer: x = 1, y = 3, z = -1 Solution: (1) 3x + y - 5z = 7 (2) x - 3y + z = -5 (3) 5x - y + 3z = 11 Multiply equation (2) by 3: 3x - 9y + 3z = -15 Subtract from equation (1): (3x + y - 5z) - (3x - 9y + 3z) = 7 - (-15) 10y - 8z = 22 Divide by 2: 5y - 4z = 11 (equation 4) Multiply equation (2) by 5: 5x - 15y + 5z = -25 Subtract from…
    Full step-by-step solution

    Step 1: Write the system: (1) 3x + y - 5z = 7 (2) x - 3y + z = -5 (3) 5x - y + 3z = 11 Step 2: Multiply equation (2) by 3: 3x - 9y + 3z = -15 Subtract from equation (1): (3x + y - 5z) - (3x - 9y + 3z) = 7 - (-15) 10y - 8z = 22 Divide by 2: 5y - 4z = 11 (equation 4) Step 3: Multiply equation (2) by 5: 5x - 15y + 5z = -25 Subtract from equation (3): (5x - y + 3z) - (5x - 15y + 5z) = 11 - (-25) 14y - 2z = 36 Divide by 2: 7y - z = 18 (equation 5) Step 4: Multiply equation (5) by 4: 28y - 4z = 72 Subtract equation (4): (28y - 4z) - (5y - 4z) = 72 - 11 23y = 61 y = 61/23 = 3 Step 5: Substitute y = 3 into equation (5): 7(3) - z = 18 21 - z = 18 z = 21 - 18 = -1 Step 6: Substitute y = 3, z = -1 into equation (2): x - 3(3) + (-1) = -5 x - 9 - 1 = -5 x - 10 = -5 x = 5 Final answer: x = 1, y = 3, z = -1

  2. x + y + z = 8, 2x - y + z = 7, x - 2y + 2z = 4 Answer: x = 3, y = 2, z = 3 Solution: (1) x + y + z = 8 (2) 2x - y + z = 7 (3) x - 2y + 2z = 4 Add equations (1) and (2) to eliminate y: (1) + (2): (x + y + z) + (2x - y + z) = 8 + 7 3x + 2z = 15 (call this equation 4) Multiply equation (1) by 2 and add to equation (3) to eliminate y: 2*(1): 2x + 2y + 2z = 16 Add to (3): (2x + 2y +…
    Full step-by-step solution

    Step 1: Label the equations: (1) x + y + z = 8 (2) 2x - y + z = 7 (3) x - 2y + 2z = 4 Step 2: Add equations (1) and (2) to eliminate y: (1) + (2): (x + y + z) + (2x - y + z) = 8 + 7 3x + 2z = 15 (call this equation 4) Step 3: Multiply equation (1) by 2 and add to equation (3) to eliminate y: 2*(1): 2x + 2y + 2z = 16 Add to (3): (2x + 2y + 2z) + (x - 2y + 2z) = 16 + 4 3x + 4z = 20 (call this equation 5) Step 4: Subtract equation (4) from equation (5) to eliminate x: (5) - (4): (3x + 4z) - (3x + 2z) = 20 - 15 2z = 5 z = 2.5 Step 5: Substitute z = 2.5 into equation (4): 3x + 2(2.5) = 15 3x + 5 = 15 3x = 10 x = 10/3 Step 6: Substitute x = 10/3 and z = 2.5 into equation (1): 10/3 + y + 2.5 = 8 y + 10/3 + 5/2 = 8 y + 20/6 + 15/6 = 8 y + 35/6 = 8 y = 8 - 35/6 y = 48/6 - 35/6 y = 13/6 The solution is x = 10/3, y = 13/6, z = 5/2.

  3. Liam is tracking his weekly earnings from three different jobs: tutoring, dog walking, and yard work. This week, he earned 3 times as much from tutoring as from dog walking, and his yard work earnings were $7 more than his dog walking earnings. The total amount he earned from all three jobs was $47. How much did Liam earn from dog walking? Answer: 8 Solution: Let x = dog walking earnings, y = tutoring earnings, z = yard work earnings.
    Full step-by-step solution

    Step 1: Let x = dog walking earnings, y = tutoring earnings, z = yard work earnings. Step 2: Translate the relationships: y = 3x (tutoring is 3 times dog walking), z = x + 7 (yard work is $7 more than dog walking), and x + y + z = 47 (total earnings). Step 3: Substitute y and z into the total equation: x + 3x + (x + 7) = 47. Step 4: Simplify: 5x + 7 = 47. Step 5: Subtract 7: 5x = 40. Step 6: Divide by 5: x = 8. The answer is 8.

  4. Solve: 3x + 5y - z = 7, x - 3y + 5z = -1, 5x + y - 3z = 11 Answer: x = 2, y = 1, z = 2 Solution: (1) 3x + 5y - z = 7 (2) x - 3y + 5z = -1 (3) 5x + y - 3z = 11 Eliminate z from equations (1) and (2): Multiply (1) by 5: 15x + 25y - 5z = 35 Add to (2): (15x + 25y - 5z) + (x - 3y + 5z) = 35 + (-1) 16x + 22y = 34 Divide by 2: 8x + 11y = 17 (Equation A) Eliminate z from equations (1) and (3):…
    Full step-by-step solution

    Step 1: Write the system: (1) 3x + 5y - z = 7 (2) x - 3y + 5z = -1 (3) 5x + y - 3z = 11 Step 2: Eliminate z from equations (1) and (2): Multiply (1) by 5: 15x + 25y - 5z = 35 Add to (2): (15x + 25y - 5z) + (x - 3y + 5z) = 35 + (-1) 16x + 22y = 34 Divide by 2: 8x + 11y = 17 (Equation A) Step 3: Eliminate z from equations (1) and (3): Multiply (1) by 3: 9x + 15y - 3z = 21 Subtract (3): (9x + 15y - 3z) - (5x + y - 3z) = 21 - 11 4x + 14y = 10 Divide by 2: 2x + 7y = 5 (Equation B) Step 4: Solve the system of equations A and B: A: 8x + 11y = 17 B: 2x + 7y = 5 Multiply B by 4: 8x + 28y = 20 Subtract A: (8x + 28y) - (8x + 11y) = 20 - 17 17y = 3 y = 3/17 Step 5: Substitute y = 3/17 into B: 2x + 7(3/17) = 5 2x + 21/17 = 5 2x = 5 - 21/17 = 85/17 - 21/17 = 64/17 x = 32/17 Step 6: Substitute x = 32/17 and y = 3/17 into equation (1): 3(32/17) + 5(3/17) - z = 7 96/17 + 15/17 - z = 7 111/17 - z = 119/17 -z = 119/17 - 111/17 = 8/17 z = -8/17 The solution is x = 32/17, y = 3/17, z = -8/17.

  5. Aroha is managing a small business that produces three types of candles: lavender, sandalwood, and vanilla. In one production run, she uses wax, fragrance oil, and wicks. For lavender candles, she uses 3 units of wax, 2 units of fragrance oil, and 1 unit of wicks. For sandalwood candles, she uses 4 units of wax, 1 unit of fragrance oil, and 2 units of wicks. For vanilla candles, she uses 2 units of wax, 3 units of fragrance oil, and 1 unit of wicks. If Aroha used 41 total units of wax, 29 total units of fragrance oil, and 18 total units of wicks, how many vanilla candles did she produce? Answer: 5 Solution: Let L = number of lavender candles, S = number of sandalwood candles, V = number of vanilla candles.
    Full step-by-step solution

    Let L = number of lavender candles, S = number of sandalwood candles, V = number of vanilla candles. Equation for wax: 3L + 4S + 2V = 41 Equation for fragrance oil: 2L + S + 3V = 29 Equation for wicks: L + 2S + V = 18 Step 1: Multiply the wicks equation by 2: 2L + 4S + 2V = 36 Step 2: Subtract this from the wax equation: (3L + 4S + 2V) - (2L + 4S + 2V) = 41 - 36 → L = 5 Step 3: Substitute L = 5 into the fragrance oil and wicks equations: Fragrance: 2(5) + S + 3V = 29 → 10 + S + 3V = 29 → S + 3V = 19 Wicks: 5 + 2S + V = 18 → 2S + V = 13 Step 4: Solve the system S + 3V = 19 and 2S + V = 13 Multiply first equation by 2: 2S + 6V = 38 Subtract second equation: (2S + 6V) - (2S + V) = 38 - 13 → 5V = 25 → V = 5 The number of vanilla candles produced is 5.

  6. 2x + 3y - z = 7, x - y + 2z = 1, 3x + y + z = 9 Answer: x = 2, y = 1, z = 2 Solution: Step 1: Label the equations: (1) 2x + 3y - z = 7 (2) x - y + 2z = 1 (3) 3x + y + z = 9 Step 2: Add equations (1) and (3) to eliminate z: (1) + (3): (2x + 3y - z) + (3x + y + z) = 7 + 9 5x + 4y = 16 (equation 4) Step 3: Multiply equation (2) by 1 and add to equation (1) to eliminate z: (1) + (2):…
    Full step-by-step solution

    Step 1: Label the equations: (1) 2x + 3y - z = 7 (2) x - y + 2z = 1 (3) 3x + y + z = 9 Step 2: Add equations (1) and (3) to eliminate z: (1) + (3): (2x + 3y - z) + (3x + y + z) = 7 + 9 5x + 4y = 16 (equation 4) Step 3: Multiply equation (2) by 1 and add to equation (1) to eliminate z: (1) + (2): (2x + 3y - z) + (x - y + 2z) = 7 + 1 3x + 2y + z = 8 (equation 5) Step 4: Now use equations (3) and (5) to eliminate z: (3) - (5): (3x + y + z) - (3x + 2y + z) = 9 - 8 -y = 1 y = -1 Step 5: Substitute y = -1 into equation (4): 5x + 4(-1) = 16 5x - 4 = 16 5x = 20 x = 4 Step 6: Substitute x = 4 and y = -1 into equation (3): 3(4) + (-1) + z = 9 12 - 1 + z = 9 11 + z = 9 z = -2 Step 7: Verify with equation (1): 2(4) + 3(-1) - (-2) = 8 - 3 + 2 = 7 ✓ Final answer: x = 4, y = -1, z = -2

  7. Solve: 3x + 2y - 4z = 7, 2x - 3y + z = -12, x + 4y - 2z = 17 Answer: x = 1, y = 4, z = 1 Solution: (1) 3x + 2y - 4z = 7 (2) 2x - 3y + z = -12 (3) x + 4y - 2z = 17 Multiply equation (2) by 4 to eliminate z with equation (1): 4*(2): 8x - 12y + 4z = -48 Add equation (1) and the modified equation (2): (3x + 2y - 4z) + (8x - 12y + 4z) = 7 + (-48) 11x - 10y = -41 (equation 4) Multiply equation (2)…
    Full step-by-step solution

    Step 1: Write the system: (1) 3x + 2y - 4z = 7 (2) 2x - 3y + z = -12 (3) x + 4y - 2z = 17 Step 2: Multiply equation (2) by 4 to eliminate z with equation (1): 4*(2): 8x - 12y + 4z = -48 Step 3: Add equation (1) and the modified equation (2): (3x + 2y - 4z) + (8x - 12y + 4z) = 7 + (-48) 11x - 10y = -41 (equation 4) Step 4: Multiply equation (2) by 2 to eliminate z with equation (3): 2*(2): 4x - 6y + 2z = -24 Step 5: Add equation (3) and the modified equation (2): (x + 4y - 2z) + (4x - 6y + 2z) = 17 + (-24) 5x - 2y = -7 (equation 5) Step 6: Now solve the system of equations (4) and (5): (4) 11x - 10y = -41 (5) 5x - 2y = -7 Step 7: Multiply equation (5) by 5: 5*(5): 25x - 10y = -35 Step 8: Subtract equation (4) from this result: (25x - 10y) - (11x - 10y) = -35 - (-41) 14x = 6 x = 6/14 = 3/7 Step 9: Substitute x = 3/7 into equation (5): 5*(3/7) - 2y = -7 15/7 - 2y = -7 -2y = -7 - 15/7 = -49/7 - 15/7 = -64/7 y = 32/7 Step 10: Substitute x = 3/7 and y = 32/7 into equation (2): 2*(3/7) - 3*(32/7) + z = -12 6/7 - 96/7 + z = -12 -90/7 + z = -12 z = -12 + 90/7 = -84/7 + 90/7 = 6/7 The solution is x = 3/7, y = 32/7, z = 6/7.

  8. Solve: 2x + y - z = 6, x - 3y + 2z = -1, 3x + 2y + z = 11 Answer: x = 3, y = 1, z = 1 Solution: When solving systems of three linear equations, the elimination method involves strategically combining equations to eliminate one variable at a time, reducing the system to two equations with two variables.
    Full step-by-step solution

    When solving systems of three linear equations, the elimination method involves strategically combining equations to eliminate one variable at a time, reducing the system to two equations with two variables. Then solve the simpler system and substitute back to find all three values.