Solving linear systems

A solution must satisfy every equation simultaneously. Use substitution or elimination to reduce a system of at most three equations in three unknowns, then substitute back into all original equations.

Example

\(\begin{aligned}x+y+z&=6\quad①\\2x-y+z&=3\quad②\\x+2y-z&=2\quad③.\end{aligned}\)

\(\begin{aligned}②-①:\quad x-2y&=-3\\③+①:\quad2x+3y&=8.\end{aligned}\)

\(\begin{aligned}x&=2y-3\\2(2y-3)+3y&=8\\7y&=14\\y&=2,\quad x=1,\quad z=3.\end{aligned}\)

Check: \(1+2+3=6\), \(2-2+3=3\) and \(1+4-3=2\).

For a technology solution, enter coefficients in a consistent variable order, including zero coefficients. A row-reduction tool uses the augmented matrix \(\left[\begin{array}{ccc|c}1&1&1&6\\2&-1&1&3\\1&2&-1&2\end{array}\right]\). Its reduced form is \(\left[\begin{array}{ccc|c}1&0&0&1\\0&1&0&2\\0&0&1&3\end{array}\right]\).