How to find original equations given the solution? Solutions to the system in Example IS are given as $$(x_1, x_2, x_3, x_4) = (−1 − 2a + 3b, 4 + a − 2b, a, b)$$ Evaluate the three equations of the original system with these expressions in $a$ and $b$ and verify that each equation is true, no matter what values are chosen for $a$ and $b$. This question was taken out of Exercise and Solution Manual for A First Course in Linear Algebra by Robert A. Beezer.

From page $13$ of _A First Course in Linear Algebra by Robert A. Beezer_, the system is given by

$$x_1 + 2x_2 + 0x_3 + x_4 = 7 \\\x_1 + x_2 + x_3 − x_4 = 3\\\ 3x_1 + x_2 + 5x_3 − 7x_4 = 1$$

Using Gaussian Elimination, we arrive at an infinite number of solutions

$$(x_1, x_2, x_3, x_4) = (−1 − 2a + 3b, 4 + a − 2b, a, b)$$

A system has infinitely many solutions when it is consistent and the number of variables is more than the number of nonzero rows in the RREF of the matrix. You should get a RREF of

$$\left(\begin{array}{rrrr|r} 1 & 0 & 2 & -3 & -1 \\\ 0 & 1 & -1 & 2 & 4 \\\ 0 & 0 & 0 & 0 & 0 \\\ \end{array}\right)$$

Can you proceed with finding that RREF, writing the stated general solution and verifying that each equation is true, no matter what values are chosen for $a$ and $b$?