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's how to solve the system of linear equations: The given system of equations is: 1) x - 2y - 2z = 0 2) 2x + 3y + z = 1 3) 3x - y - 3z = 3 Step 1: Eliminate z from equations (1) and (2). Multiply equation (2) by 2: 2(2x + 3y + z) = 2(1) 4x + 6y + 2z = 2 (Equation 4) Add equation (1) and equation (4): (x - 2y - 2z) + (4x + 6y + 2z) = 0 + 2 5x + 4y = 2 (Equation A) Step 2: Eliminate z from equations (2) and (3). Multiply equation (2) by 3: 3(2x + 3y + z) = 3(1) 6x + 9y + 3z = 3 (Equation 5) Add equation (3) and equation (5): (3x - y - 3z) + (6x + 9y + 3z) = 3 + 3 9x + 8y = 6 (Equation B) Step 3: Solve the system of two equations (A and B) for x and y. Equation A: 5x + 4y = 2 Equation B: 9x + 8y = 6 Multiply equation (A) by 2: 2(5x + 4y) = 2(2) 10x + 8y = 4 (Equation C) Subtract equation (C) from equation (B): (9x + 8y) - (10x + 8y) = 6 - 4 -x = 2 x = -2 Step 4: Substitute the value of x into equation (A) to find y. 5(-2) + 4y = 2 -10 + 4y = 2 4y = 2 + 10 4y = 12 y = (12)/(4) y = 3 Step 5: Substitute the values of x and y into equation (1) to find z. (-2) - 2(3) - 2z = 0 -2 - 6 - 2z = 0 -8 - 2z = 0 -2z = 8 z = (8)/(-2) z = -4 Step 6: Verify the solution with the original equations. For (1): (-2) - 2(3) - 2(-4) = -2 - 6 + 8 = 0 (Correct) For (2): 2(-2) + 3(3) + (-4) = -4 + 9 - 4 = 1 (Correct) For (3): 3(-2) - (3) - 3(-4) = -6 - 3 + 12 = 3 (Correct) The solution is x=-2, y=3, and z=-4. x=-2, y=3, z=-4 That's 2 down. 3 left today — send the next one.