Linear Systems 3x3

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Linear Systems 3x3 · Grade 12 · Worksheet 1

1. x + 3y - z = 6, 2x - y + 4z = 1, 3x + 2y - z = 11 Answer: x = 3, y = 1, z = 0 Solution: Start with the equations: (1) x + 3y - z = 6 (2) 2x - y + 4z = 1 (3) 3x + 2y - z = 11 Multiply equation (1) by 2: 2x + 6y - 2z = 12 Subtract equation (2): (2x + 6y - 2z) - (2x - y + 4z) = 12 - 1 This gives: 7y - 6z = 11 (Equation A) Multiply equation (1) by 3: 3x + 9y - 3z = 18 Subtract…
   Full step-by-step solution

   Step 1: Let's use elimination. Start with the equations: (1) x + 3y - z = 6 (2) 2x - y + 4z = 1 (3) 3x + 2y - z = 11 Step 2: Multiply equation (1) by 2: 2x + 6y - 2z = 12 Subtract equation (2): (2x + 6y - 2z) - (2x - y + 4z) = 12 - 1 This gives: 7y - 6z = 11 (Equation A) Step 3: Multiply equation (1) by 3: 3x + 9y - 3z = 18 Subtract equation (3): (3x + 9y - 3z) - (3x + 2y - z) = 18 - 11 This gives: 7y - 2z = 7 (Equation B) Step 4: Subtract Equation B from Equation A: (7y - 6z) - (7y - 2z) = 11 - 7 This gives: -4z = 4, so z = -1 Step 5: Substitute z = -1 into Equation B: 7y - 2(-1) = 7 7y + 2 = 7 7y = 5 y = 5/7 Step 6: Substitute y = 5/7 and z = -1 into equation (1): x + 3(5/7) - (-1) = 6 x + 15/7 + 1 = 6 x + 15/7 + 7/7 = 6 x + 22/7 = 42/7 x = 42/7 - 22/7 x = 20/7 The solution is x = 20/7, y = 5/7, z = -1.

2. 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

3. Solve: 2x + y - z = 11, x - 2y + 3z = 9, 3x + y + 2z = 18 Answer: x = 4, y = 3, z = 2 Solution: (1) 2x + y - z = 11 (2) x - 2y + 3z = 9 (3) 3x + y + 2z = 18 Eliminate y from equations (1) and (3): Subtract (1) from (3): (3x + y + 2z) - (2x + y - z) = 18 - 11 x + 3z = 7 → (4) Eliminate y from equations (1) and (2): Multiply (1) by 2: 4x + 2y - 2z = 22 Add to (2): (4x + 2y - 2z) + (x - 2y…
   Full step-by-step solution

   Step 1: Label the equations: (1) 2x + y - z = 11 (2) x - 2y + 3z = 9 (3) 3x + y + 2z = 18 Step 2: Eliminate y from equations (1) and (3): Subtract (1) from (3): (3x + y + 2z) - (2x + y - z) = 18 - 11 x + 3z = 7 → (4) Step 3: Eliminate y from equations (1) and (2): Multiply (1) by 2: 4x + 2y - 2z = 22 Add to (2): (4x + 2y - 2z) + (x - 2y + 3z) = 22 + 9 5x + z = 31 → (5) Step 4: Solve equations (4) and (5): From (4): x = 7 - 3z Substitute into (5): 5(7 - 3z) + z = 31 35 - 15z + z = 31 35 - 14z = 31 -14z = -4 z = 2/7 Step 5: Find x: x = 7 - 3(2/7) = 7 - 6/7 = 49/7 - 6/7 = 43/7 Step 6: Find y using equation (1): 2(43/7) + y - 2/7 = 11 86/7 + y - 2/7 = 11 84/7 + y = 11 12 + y = 11 y = -1 Step 7: Check with equation (2): 43/7 - 2(-1) + 3(2/7) = 43/7 + 2 + 6/7 = 49/7 + 2 = 7 + 2 = 9 ✓ Step 8: Check with equation (3): 3(43/7) + (-1) + 2(2/7) = 129/7 - 1 + 4/7 = 133/7 - 1 = 19 - 1 = 18 ✓ The solution is x = 43/7, y = -1, z = 2/7.

4. Solve: 4x + 3y - 2z = 12, 2x - 5y + z = -8, x + 2y + 3z = 10 Answer: x = 2, y = 2, z = 1 Solution: To solve a system of three linear equations with three variables, you can use elimination or substitution methods. This method works because each equation represents a plane in 3D space, and the solution is the point where all three planes intersect.
   Full step-by-step solution

   To solve a system of three linear equations with three variables, you can use elimination or substitution methods. The goal is to eliminate one variable at a time to reduce the system to two equations with two variables, then solve that simpler system. Once you find two variables, substitute back to find the third. This method works because each equation represents a plane in 3D space, and the solution is the point where all three planes intersect.

5. Solve: 3x + 5y - z = 11, x - 3y + 2z = -7, 5x + y - 3z = 13 Answer: x = 3, y = 1, z = 3 Solution: (1) 3x + 5y - z = 11 (2) x - 3y + 2z = -7 (3) 5x + y - 3z = 13 Multiply equation (1) by 2: 6x + 10y - 2z = 22 Add this to equation (2): (x - 3y + 2z) + (6x + 10y - 2z) = -7 + 22 7x + 7y = 15 (equation 4) Multiply equation (1) by -3: -9x - 15y + 3z = -33 Add this to equation (3): (5x + y - 3z) +…
   Full step-by-step solution

   Step 1: Label the equations: (1) 3x + 5y - z = 11 (2) x - 3y + 2z = -7 (3) 5x + y - 3z = 13 Step 2: Multiply equation (1) by 2: 6x + 10y - 2z = 22 Add this to equation (2): (x - 3y + 2z) + (6x + 10y - 2z) = -7 + 22 7x + 7y = 15 (equation 4) Step 3: Multiply equation (1) by -3: -9x - 15y + 3z = -33 Add this to equation (3): (5x + y - 3z) + (-9x - 15y + 3z) = 13 + (-33) -4x - 14y = -20 (equation 5) Step 4: Simplify equation (5) by dividing by -2: 2x + 7y = 10 Step 5: Now solve the system of equations (4) and (5): (4) 7x + 7y = 15 (5) 2x + 7y = 10 Step 6: Subtract equation (5) from equation (4): (7x - 2x) + (7y - 7y) = 15 - 10 5x = 5 x = 1 Step 7: Substitute x = 1 into equation (5): 2(1) + 7y = 10 2 + 7y = 10 7y = 8 y = 8/7 Step 8: Substitute x = 1 and y = 8/7 into equation (1): 3(1) + 5(8/7) - z = 11 3 + 40/7 - z = 11 21/7 + 40/7 - z = 11 61/7 - z = 11 -z = 11 - 61/7 = 77/7 - 61/7 = 16/7 z = -16/7 The solution is x = 1, y = 8/7, z = -16/7.

6. 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.

7. Isabella invests in three stocks. Stock A costs $8 per share, Stock B costs $9 per share, and Stock C costs $10 per share. She buys 4 shares of A, 3 shares of B, and 2 shares of C for a total of $80. She also buys 2 shares of A, 5 shares of B, and 1 share of C for $75. Additionally, she buys 3 shares of A, 1 share of B, and 4 shares of C for $70. Find the price per share for each stock. Answer: A = 5, B = 8, C = 6 Solution: Let A, B, and C be the prices per share of stocks A, B, and C respectively. Equation 1 (first purchase): 4A + 3B + 2C = 80 Equation 2 (second purchase): 2A + 5B + C = 75 Equation 3 (third purchase): 3A + B + 4C = 70 Use Equation 2 to express C in terms of A and B: C = 75 - 2A - 5B Substitute C…
   Full step-by-step solution

   Let A, B, and C be the prices per share of stocks A, B, and C respectively. Equation 1 (first purchase): 4A + 3B + 2C = 80 Equation 2 (second purchase): 2A + 5B + C = 75 Equation 3 (third purchase): 3A + B + 4C = 70 Step 1: Use Equation 2 to express C in terms of A and B: C = 75 - 2A - 5B Step 2: Substitute C into Equation 1: 4A + 3B + 2(75 - 2A - 5B) = 80 4A + 3B + 150 - 4A - 10B = 80 -7B + 150 = 80 -7B = -70 B = 10 Step 3: Substitute B = 10 into Equation 2: 2A + 5(10) + C = 75 2A + 50 + C = 75 2A + C = 25 C = 25 - 2A Step 4: Substitute B = 10 and C = 25 - 2A into Equation 3: 3A + 10 + 4(25 - 2A) = 70 3A + 10 + 100 - 8A = 70 -5A + 110 = 70 -5A = -40 A = 8 Step 5: Substitute A = 8 into C = 25 - 2A: C = 25 - 2(8) = 25 - 16 = 9 Therefore, Stock A costs $8 per share, Stock B costs $10 per share, and Stock C costs $9 per share.

8. 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.

9. 2x + y - z = 5, x - 2y + z = -5, 3x + y + 2z = 10 Answer: x = 2, y = 1, z = 0 Solution: Step 1: Label the equations: (1) 2x + y - z = 5 (2) x - 2y + z = -5 (3) 3x + y + 2z = 10 Step 2: Add equations (1) and (2) to eliminate z: (1) + (2): (2x + x) + (y - 2y) + (-z + z) = 5 + (-5) 3x - y = 0 So y = 3x Step 3: Substitute y = 3x into equation (1): 2x + (3x) - z = 5 5x - z = 5 So z = 5x…
   Full step-by-step solution

   Step 1: Label the equations: (1) 2x + y - z = 5 (2) x - 2y + z = -5 (3) 3x + y + 2z = 10 Step 2: Add equations (1) and (2) to eliminate z: (1) + (2): (2x + x) + (y - 2y) + (-z + z) = 5 + (-5) 3x - y = 0 So y = 3x Step 3: Substitute y = 3x into equation (1): 2x + (3x) - z = 5 5x - z = 5 So z = 5x - 5 Step 4: Substitute y = 3x and z = 5x - 5 into equation (3): 3x + (3x) + 2(5x - 5) = 10 3x + 3x + 10x - 10 = 10 16x - 10 = 10 16x = 20 x = 20/16 = 5/4 Step 5: Find y and z: y = 3x = 3(5/4) = 15/4 z = 5x - 5 = 5(5/4) - 5 = 25/4 - 20/4 = 5/4 Step 6: Verify with equation (3): 3(5/4) + 15/4 + 2(5/4) = 15/4 + 15/4 + 10/4 = 40/4 = 10 ✓ The solution is x = 5/4, y = 15/4, z = 5/4.