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
Step 1: Write down the given system of linear equations. The system of equations is: 1. x+y+z=p 2. 2x-y+3z=q 3. 4x+y+5z=r Step 2: Perform a linear combination of the first two equations. Multiply equation (1) by 2 and add it to equation (2): 2 × (x+y+z) + (2x-y+3z) = 2p + q 2x+2y+2z + 2x-y+3z = 2p+q 4x+y+5z = 2p+q Step 3: Compare the result with the third equation. We have derived the equation 4x+y+5z = 2p+q. The third equation given in the system is 4x+y+5z = r. Step 4: Conclude inconsistency. If the system is to have a solution, then the derived equation must be consistent with the third equation. This means that the right-hand sides must be equal: r = 2p+q However, the problem states that the system is inconsistent if r ≠ 2p+q. If r ≠ 2p+q, then we have a contradiction: 4x+y+5z = 2p+q 4x+y+5z = r Since 2p+q ≠ r, the same expression 4x+y+5z cannot simultaneously be equal to two different values, 2p+q and r. Therefore, no values of x, y, z can satisfy all three equations simultaneously. Thus, the system of equations is inconsistent if r ≠ 2p+q. That's 2 down. 3 left today — send the next one.