Linear Systems 3x3

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Linear Systems 3x3 · Grade 12 · Worksheet 3

1. Solve: 2x + 3y - 2z = 7, x - 2y + 4z = -2, 3x + y - z = 12 Answer: x = 3, y = 2, z = 1/2 Solution: (1) 2x + 3y - 2z = 7 (2) x - 2y + 4z = -2 (3) 3x + y - z = 12 Multiply equation (3) by 3: 9x + 3y - 3z = 36 Subtract equation (1) from this: (9x + 3y - 3z) - (2x + 3y - 2z) = 36 - 7 7x - z = 29 (equation 4) Multiply equation (3) by 2: 6x + 2y - 2z = 24 Add this to equation (2): (x - 2y + 4z) +…
   Full step-by-step solution

   Step 1: Write the system: (1) 2x + 3y - 2z = 7 (2) x - 2y + 4z = -2 (3) 3x + y - z = 12 Step 2: Multiply equation (3) by 3: 9x + 3y - 3z = 36 Subtract equation (1) from this: (9x + 3y - 3z) - (2x + 3y - 2z) = 36 - 7 7x - z = 29 (equation 4) Step 3: Multiply equation (3) by 2: 6x + 2y - 2z = 24 Add this to equation (2): (x - 2y + 4z) + (6x + 2y - 2z) = -2 + 24 7x + 2z = 22 (equation 5) Step 4: Now solve the system of equations (4) and (5): (4) 7x - z = 29 (5) 7x + 2z = 22 Step 5: Subtract equation (4) from equation (5): (7x + 2z) - (7x - z) = 22 - 29 3z = -7 z = -7/3 Step 6: Substitute z = -7/3 into equation (4): 7x - (-7/3) = 29 7x + 7/3 = 29 7x = 29 - 7/3 = 87/3 - 7/3 = 80/3 x = 80/21 Step 7: Substitute x = 80/21 and z = -7/3 into equation (3): 3(80/21) + y - (-7/3) = 12 240/21 + y + 7/3 = 12 80/7 + y + 7/3 = 12 y = 12 - 80/7 - 7/3 y = 252/21 - 240/21 - 49/21 = -37/21 The solution is x = 80/21, y = -37/21, z = -7/3.

2. Olivia is mixing fruit juices for a party. She has three types: apple, orange, and pineapple. The total volume of juice is 15 liters. The apple juice volume plus twice the orange juice volume minus the pineapple juice volume equals 5 liters. Also, twice the apple juice volume plus the orange juice volume plus the pineapple juice volume equals 20 liters. How many liters of apple juice does Olivia have? Answer: 5 Solution: Let x = liters of apple juice, y = liters of orange juice, z = liters of pineapple juice.
   Full step-by-step solution

   Let x = liters of apple juice, y = liters of orange juice, z = liters of pineapple juice. Equation 1: x + y + z = 15 Equation 2: x + 2y - z = 5 Equation 3: 2x + y + z = 20 Step 1: Add Equation 1 and Equation 2: (x + y + z) + (x + 2y - z) = 15 + 5 2x + 3y = 20 Step 2: Add Equation 1 and Equation 3: (x + y + z) + (2x + y + z) = 15 + 20 3x + 2y + 2z = 35 Step 3: Subtract Equation 2 from Equation 3: (2x + y + z) - (x + 2y - z) = 20 - 5 x - y + 2z = 15 Step 4: From Step 1: 2x + 3y = 20 From Step 3: x - y + 2z = 15 Step 5: Use Equation 1: x + y + z = 15 Multiply by 2: 2x + 2y + 2z = 30 Step 6: Subtract Step 3 from this result: (2x + 2y + 2z) - (x - y + 2z) = 30 - 15 x + 3y = 15 Step 7: Now we have: 2x + 3y = 20 x + 3y = 15 Step 8: Subtract the second from the first: (2x + 3y) - (x + 3y) = 20 - 15 x = 5 Olivia has 5 liters of apple juice.

3. Solve: 2x + 3y - z = 12, x - 2y + 3z = 9, 3x + y - 2z = 11 Answer: x = 4, y = 2, z = 3 Solution: (1) 2x + 3y - z = 12 (2) x - 2y + 3z = 9 (3) 3x + y - 2z = 11 Eliminate z from equations (1) and (2): Multiply (1) by 3: 6x + 9y - 3z = 36 Add to (2): (6x + 9y - 3z) + (x - 2y + 3z) = 36 + 9 7x + 7y = 45 → (4) x + y = 45/7 Eliminate z from equations (1) and (3): Multiply (1) by 2: 4x + 6y - 2z…
   Full step-by-step solution

   Step 1: Label the equations: (1) 2x + 3y - z = 12 (2) x - 2y + 3z = 9 (3) 3x + y - 2z = 11 Step 2: Eliminate z from equations (1) and (2): Multiply (1) by 3: 6x + 9y - 3z = 36 Add to (2): (6x + 9y - 3z) + (x - 2y + 3z) = 36 + 9 7x + 7y = 45 → (4) x + y = 45/7 Step 3: Eliminate z from equations (1) and (3): Multiply (1) by 2: 4x + 6y - 2z = 24 Subtract (3): (4x + 6y - 2z) - (3x + y - 2z) = 24 - 11 x + 5y = 13 → (5) Step 4: Solve equations (4) and (5): From (4): x = 45/7 - y Substitute into (5): (45/7 - y) + 5y = 13 45/7 + 4y = 13 4y = 13 - 45/7 = 91/7 - 45/7 = 46/7 y = 46/28 = 23/14 Step 5: Find x: x = 45/7 - 23/14 = 90/14 - 23/14 = 67/14 Step 6: Find z using equation (1): 2(67/14) + 3(23/14) - z = 12 134/14 + 69/14 - z = 12 203/14 - z = 12 z = 203/14 - 168/14 = 35/14 = 5/2 Step 7: Check with equation (2): (67/14) - 2(23/14) + 3(5/2) = 67/14 - 46/14 + 15/2 = 21/14 + 105/14 = 126/14 = 9 ✓ Step 8: Check with equation (3): 3(67/14) + (23/14) - 2(5/2) = 201/14 + 23/14 - 5 = 224/14 - 70/14 = 154/14 = 11 ✓ The solution is x = 67/14, y = 23/14, z = 5/2.

4. Ava is designing a triangular garden plot with sides represented by the equations: 3x + 2y - z = 14, x - 4y + 2z = -3, and 2x + y + 3z = 19. The side lengths (in meters) correspond to the values of x, y, and z. What is the perimeter of the triangular garden? Answer: 15 Solution: 3x + 2y - z = 14 (Equation 1) x - 4y + 2z = -3 (Equation 2) 2x + y + 3z = 19 (Equation 3) Multiply Equation 1 by 2 and add to Equation 2: 2(3x + 2y - z) = 2(14) → 6x + 4y - 2z = 28 (6x + 4y - 2z) + (x - 4y + 2z) = 28 + (-3) 7x = 25 x = 25/7 Multiply Equation 1 by 3 and add to Equation 3: 3(3x +…
   Full step-by-step solution

   Step 1: Write the system of equations: 3x + 2y - z = 14 (Equation 1) x - 4y + 2z = -3 (Equation 2) 2x + y + 3z = 19 (Equation 3) Step 2: Multiply Equation 1 by 2 and add to Equation 2: 2(3x + 2y - z) = 2(14) → 6x + 4y - 2z = 28 (6x + 4y - 2z) + (x - 4y + 2z) = 28 + (-3) 7x = 25 x = 25/7 Step 3: Multiply Equation 1 by 3 and add to Equation 3: 3(3x + 2y - z) = 3(14) → 9x + 6y - 3z = 42 (9x + 6y - 3z) + (2x + y + 3z) = 42 + 19 11x + 7y = 61 Step 4: Substitute x = 25/7 into 11x + 7y = 61: 11(25/7) + 7y = 61 275/7 + 7y = 61 7y = 61 - 275/7 = 427/7 - 275/7 = 152/7 y = 152/49 Step 5: Substitute x = 25/7 and y = 152/49 into Equation 1: 3(25/7) + 2(152/49) - z = 14 75/7 + 304/49 - z = 14 525/49 + 304/49 - z = 14 829/49 - z = 14 z = 829/49 - 686/49 = 143/49 Step 6: Calculate the perimeter (x + y + z): x + y + z = 25/7 + 152/49 + 143/49 = 175/49 + 152/49 + 143/49 = 470/49 = 15 The perimeter of the triangular garden is 15 meters.

5. 2x + y - z = 5, x - 3y + 2z = -5, 3x + 2y + z = 10 Answer: x = 3, y = 1, z = 2 Solution: Step 1: Label the equations: (1) 2x + y - z = 5 (2) x - 3y + 2z = -5 (3) 3x + 2y + z = 10 Step 2: Add equations (1) and (3) to eliminate z: (1) + (3): (2x + y - z) + (3x + 2y + z) = 5 + 10 5x + 3y = 15 (equation 4) Step 3: Multiply equation (1) by 2 and add to equation (2): 2*(1): 4x + 2y - 2z =…
   Full step-by-step solution

   Step 1: Label the equations: (1) 2x + y - z = 5 (2) x - 3y + 2z = -5 (3) 3x + 2y + z = 10 Step 2: Add equations (1) and (3) to eliminate z: (1) + (3): (2x + y - z) + (3x + 2y + z) = 5 + 10 5x + 3y = 15 (equation 4) Step 3: Multiply equation (1) by 2 and add to equation (2): 2*(1): 4x + 2y - 2z = 10 Add to (2): (4x + 2y - 2z) + (x - 3y + 2z) = 10 + (-5) 5x - y = 5 (equation 5) Step 4: Solve the system of equations (4) and (5): (4) 5x + 3y = 15 (5) 5x - y = 5 Subtract (5) from (4): (5x + 3y) - (5x - y) = 15 - 5 4y = 10 y = 2.5 Step 5: Substitute y = 2.5 into equation (5): 5x - 2.5 = 5 5x = 7.5 x = 1.5 Step 6: Substitute x = 1.5 and y = 2.5 into equation (1): 2(1.5) + 2.5 - z = 5 3 + 2.5 - z = 5 5.5 - z = 5 z = 0.5 Step 7: Verify with equation (3): 3(1.5) + 2(2.5) + 0.5 = 4.5 + 5 + 0.5 = 10 ✓ The solution is x = 1.5, y = 2.5, z = 0.5

6. Solve: 2x + y - z = 8, x - 2y + 3z = 7, 3x + y + 2z = 17 Answer: x = 3, y = 2, z = 1 Solution: When solving systems of three linear equations, the elimination method involves strategically adding or subtracting equations to eliminate one variable, 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 adding or subtracting equations to eliminate one variable, reducing the system to two equations with two variables. Once you solve for two variables, substitute back to find the third.

7. Sophia is mixing three chemical solutions. Solution A contains 4x + 3y - 2z liters, Solution B contains 2x - 4y + 5z liters, and Solution C contains 3x + 2y - z liters. If she needs 15 liters of Solution A, 8 liters of Solution B, and 12 liters of Solution C, find x, y, and z: 4x + 3y - 2z = 15, 2x - 4y + 5z = 8, 3x + 2y - z = 12 Answer: x = 3, y = 2, z = 1 Solution: (1) 4x + 3y - 2z = 15 (2) 2x - 4y + 5z = 8 (3) 3x + 2y - z = 12 Multiply equation (3) by 2: 6x + 4y - 2z = 24 Subtract equation (1) from this: (6x + 4y - 2z) - (4x + 3y - 2z) = 24 - 15 2x + y = 9 (equation 4) Multiply equation (3) by 5: 15x + 10y - 5z = 60 Add this to equation (2): (2x - 4y +…
   Full step-by-step solution

   Step 1: Write the system: (1) 4x + 3y - 2z = 15 (2) 2x - 4y + 5z = 8 (3) 3x + 2y - z = 12 Step 2: Multiply equation (3) by 2: 6x + 4y - 2z = 24 Subtract equation (1) from this: (6x + 4y - 2z) - (4x + 3y - 2z) = 24 - 15 2x + y = 9 (equation 4) Step 3: Multiply equation (3) by 5: 15x + 10y - 5z = 60 Add this to equation (2): (2x - 4y + 5z) + (15x + 10y - 5z) = 8 + 60 17x + 6y = 68 (equation 5) Step 4: Multiply equation (4) by 6: 12x + 6y = 54 Subtract this from equation (5): (17x + 6y) - (12x + 6y) = 68 - 54 5x = 14 x = 14/5 = 2.8 Step 5: Substitute x = 14/5 into equation (4): 2(14/5) + y = 9 28/5 + y = 9 y = 9 - 28/5 = 45/5 - 28/5 = 17/5 = 3.4 Step 6: Substitute x = 14/5 and y = 17/5 into equation (3): 3(14/5) + 2(17/5) - z = 12 42/5 + 34/5 - z = 12 76/5 - z = 12 z = 76/5 - 12 = 76/5 - 60/5 = 16/5 = 3.2 The solution is x = 14/5, y = 17/5, z = 16/5.

8. Solve: 2x + 3y - z = 15, x - 2y + 2z = 7, 3x + y - 2z = 16 Answer: x = 5, y = 3, z = 4 Solution: Step 1: Label the equations: (1) 2x + 3y - z = 15 (2) x - 2y + 2z = 7 (3) 3x + y - 2z = 16 Step 2: Eliminate z from equations (1) and (3): Multiply (1) by 2: 4x + 6y - 2z = 30 Subtract (3): (4x + 6y - 2z) - (3x + y - 2z) = 30 - 16 x + 5y = 14 (equation 4) Step 3: Eliminate z from equations (1)…
   Full step-by-step solution

   Step 1: Label the equations: (1) 2x + 3y - z = 15 (2) x - 2y + 2z = 7 (3) 3x + y - 2z = 16 Step 2: Eliminate z from equations (1) and (3): Multiply (1) by 2: 4x + 6y - 2z = 30 Subtract (3): (4x + 6y - 2z) - (3x + y - 2z) = 30 - 16 x + 5y = 14 (equation 4) Step 3: Eliminate z from equations (1) and (2): Multiply (1) by 2: 4x + 6y - 2z = 30 Add to (2): (4x + 6y - 2z) + (x - 2y + 2z) = 30 + 7 5x + 4y = 37 (equation 5) Step 4: Solve the system of equations (4) and (5): (4) x + 5y = 14 (5) 5x + 4y = 37 Multiply (4) by 5: 5x + 25y = 70 Subtract (5): (5x + 25y) - (5x + 4y) = 70 - 37 21y = 33 y = 33/21 = 11/7 Step 5: Substitute y = 11/7 into equation (4): x + 5(11/7) = 14 x + 55/7 = 14 x = 14 - 55/7 = 98/7 - 55/7 = 43/7 Step 6: Substitute x = 43/7 and y = 11/7 into equation (1): 2(43/7) + 3(11/7) - z = 15 86/7 + 33/7 - z = 15 119/7 - z = 15 17 - z = 15 z = 2 Step 7: Verify with equation (2): (43/7) - 2(11/7) + 2(2) = 43/7 - 22/7 + 4 = 21/7 + 4 = 3 + 4 = 7 ✓ Step 8: Verify with equation (3): 3(43/7) + (11/7) - 2(2) = 129/7 + 11/7 - 4 = 140/7 - 4 = 20 - 4 = 16 ✓ The solution is x = 43/7, y = 11/7, z = 2.