Solve {lll}{1}&{2}&{3}{4}&{5}&{6}{7}&{8}&{5}{l}{x}{y}{z}={l}{5}{10}{15}

$$\left( \begin{array} { l l l } { 1 } & { 2 } & { 3 } \\ { 4 } & { 5 } & { 6 } \\ { 7 } & { 8 } & { 5 } \end{array} \right) \left( \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right) = \left( \begin{array} { l } { 5 } \\ { 10 } \\ { 15 } \end{array} \right)$$

Solve for x, y, z

$$x = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

$$y = \frac{10}{3} = 3\frac{1}{3} \approx 3.333333333$$

$$z=0$$

Short Steps Using Substitution

Solve x+2y+3z=5 for x.

$$x=-2y-3z+5$$

Substitute -2y-3z+5 for x in the second and third equation.

$$4\left(-2y-3z+5\right)+5y+6z=10 7\left(-2y-3z+5\right)+8y+5z=15$$

Solve these equations for y and z respectively.

$$y=\frac{10}{3}-2z z=\frac{5}{4}-\frac{3}{8}y$$

Substitute \frac{10}{3}-2z for y in the equation z=\frac{5}{4}-\frac{3}{8}y.

$$z=\frac{5}{4}-\frac{3}{8}\left(\frac{10}{3}-2z\right)$$

Solve z=\frac{5}{4}-\frac{3}{8}\left(\frac{10}{3}-2z\right) for z.

$$z=0$$

Substitute 0 for z in the equation y=\frac{10}{3}-2z.

$$y=\frac{10}{3}-2\times 0$$

Calculate y from y=\frac{10}{3}-2\times 0.

$$y=\frac{10}{3}$$

Substitute \frac{10}{3} for y and 0 for z in the equation x=-2y-3z+5.

$$x=-2\times \frac{10}{3}-3\times 0+5$$

Calculate x from x=-2\times \frac{10}{3}-3\times 0+5.

$$x=-\frac{5}{3}$$

The system is now solved.

$$x=-\frac{5}{3} y=\frac{10}{3} z=0$$