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

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

Solve for x, y, z

$$x=2$$

$$y=-3$$

$$z=\frac{1}{2}=0.5$$

Short Steps Using Substitution

Reorder the equations.

$$y=2x-7 3x-4z=4 5x+3y+2z=2$$

Substitute 2x-7 for y in the equation 5x+3y+2z=2.

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

Solve the second equation for x and the third equation for z.

$$x=\frac{4}{3}+\frac{4}{3}z z=-\frac{11}{2}x+\frac{23}{2}$$

Substitute \frac{4}{3}+\frac{4}{3}z for x in the equation z=-\frac{11}{2}x+\frac{23}{2}.

$$z=-\frac{11}{2}\left(\frac{4}{3}+\frac{4}{3}z\right)+\frac{23}{2}$$

Solve z=-\frac{11}{2}\left(\frac{4}{3}+\frac{4}{3}z\right)+\frac{23}{2} for z.

$$z=\frac{1}{2}$$

Substitute \frac{1}{2} for z in the equation x=\frac{4}{3}+\frac{4}{3}z.

$$x=\frac{4}{3}+\frac{4}{3}\times \frac{1}{2}$$

Calculate x from x=\frac{4}{3}+\frac{4}{3}\times \frac{1}{2}.

$$x=2$$

Substitute 2 for x in the equation y=2x-7.

$$y=2\times 2-7$$

Calculate y from y=2\times 2-7.

$$y=-3$$

The system is now solved.

$$x=2 y=-3 z=\frac{1}{2}$$