Solve {l}{2x+3y-z=7}{3x-3y+2z=2}{4x+5y-4z=3}

$$\left\{ \begin{array} { l } { 2 x + 3 y - z = 7 } \\ { 3 x - 3 y + 2 z = 2 } \\ { 4 x + 5 y - 4 z = 3 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=3$$

$$z=4$$

Short Steps Using Substitution

Solve 2x+3y-z=7 for z.

$$z=2x+3y-7$$

Substitute 2x+3y-7 for z in the second and third equation.

$$3x-3y+2\left(2x+3y-7\right)=2 4x+5y-4\left(2x+3y-7\right)=3$$

Solve these equations for y and x respectively.

$$y=-\frac{7}{3}x+\frac{16}{3} x=-\frac{7}{4}y+\frac{25}{4}$$

Substitute -\frac{7}{3}x+\frac{16}{3} for y in the equation x=-\frac{7}{4}y+\frac{25}{4}.

$$x=-\frac{7}{4}\left(-\frac{7}{3}x+\frac{16}{3}\right)+\frac{25}{4}$$

Solve x=-\frac{7}{4}\left(-\frac{7}{3}x+\frac{16}{3}\right)+\frac{25}{4} for x.

$$x=1$$

Substitute 1 for x in the equation y=-\frac{7}{3}x+\frac{16}{3}.

$$y=-\frac{7}{3}+\frac{16}{3}$$

Calculate y from y=-\frac{7}{3}+\frac{16}{3}.

$$y=3$$

Substitute 3 for y and 1 for x in the equation z=2x+3y-7.

$$z=2\times 1+3\times 3-7$$

Calculate z from z=2\times 1+3\times 3-7.

$$z=4$$

The system is now solved.

$$x=1 y=3 z=4$$