This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the steps to solve the system of linear equations: The given system of equations is: 1) 4x_1 + x_2 - 5x_3 = 8 2) -2x_1 + 3x_2 + x_3 = 12 3) 3x_1 + x_2 + 4x_3 = 5 Step 1: Eliminate x_1 from equations (1) and (2). Multiply equation (2) by 2: 2(-2x_1 + 3x_2 + x_3) = 2(12) -4x_1 + 6x_2 + 2x_3 = 24 (Equation 4) Add Equation (1) to Equation (4): (4x_1 + x_2 - 5x_3) + (-4x_1 + 6x_2 + 2x_3) = 8 + 24 7x_2 - 3x_3 = 32 (Equation 5) Step 2: Eliminate x_1 from equations (1) and (3). Multiply equation (1) by 3: 3(4x_1 + x_2 - 5x_3) = 3(8) 12x_1 + 3x_2 - 15x_3 = 24 (Equation 6) Multiply equation (3) by 4: 4(3x_1 + x_2 + 4x_3) = 4(5) 12x_1 + 4x_2 + 16x_3 = 20 (Equation 7) Subtract Equation (6) from Equation (7): (12x_1 + 4x_2 + 16x_3) - (12x_1 + 3x_2 - 15x_3) = 20 - 24 12x_1 + 4x_2 + 16x_3 - 12x_1 - 3x_2 + 15x_3 = -4 x_2 + 31x_3 = -4 (Equation 8) Step 3: Solve the system of equations (5) and (8) for x_2 and x_3. The new system is: 5) 7x_2 - 3x_3 = 32 8) x_2 + 31x_3 = -4 From Equation (8), express x_2 in terms of x_3: x_2 = -4 - 31x_3 Substitute this expression for x_2 into Equation (5): 7(-4 - 31x_3) - 3x_3 = 32 -28 - 217x_3 - 3x_3 = 32 -220x_3 = 32 + 28 -220x_3 = 60 x_3 = -(60)/(220) x_3 = -(3)/(11) Step 4: Substitute the value of x_3 into Equation (8) to find x_2. x_2 = -4 - 31(-(3)/(11)) x_2 = -4 + (93)/(11) x_2 = (-4 × 11 + 93)/(11) x_2 = (-44 + 93)/(11) x_2 = (49)/(11) Step 5: Substitute the values of x_2 and x_3 into Equation (1) to find x_1. 4x_1 + x_2 - 5x_3 = 8 4x_1 + (49)/(11) - 5(-(3)/(11)) = 8 4x_1 + (49)/(11) + (15)/(11) = 8 4x_1 + (49 + 15)/(11) = 8 4x_1 + (64)/(11) = 8 4x_1 = 8 - (64)/(11) 4x_1 = (8 × 11 - 64)/(11) 4x_1 = (88 - 64)/(11) 4x_1 = (24)/(11) x_1 = (24)/(4 × 11) x_1 = (6)/(11) The solution to the system of equations is: x_1 = (6)/(11), x_2 = (49)/(11), x_3 = -(3)/(11) x_1 = (6)/(11), x_2 = (49)/(11), x_3 = -(3)/(11) 3 done, 2 left today. You're making progress.