Solve {l}{3x-4y+2z=5}{5y-3z=-12}{7x+2z=1}

$$\left. \begin{array}{l}{ 3 x - 4 y + 2 z = 5 }\\{ 5 y - 3 z = - 12 }\\{ 7 x + 2 z = 1 }\end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=0$$

$$z=4$$

Short Steps Using Substitution

Solve 3x-4y+2z=5 for x.

$$x=\frac{4}{3}y-\frac{2}{3}z+\frac{5}{3}$$

Substitute \frac{4}{3}y-\frac{2}{3}z+\frac{5}{3} for x in the equation 7x+2z=1.

$$7\left(\frac{4}{3}y-\frac{2}{3}z+\frac{5}{3}\right)+2z=1$$

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

$$y=-\frac{12}{5}+\frac{3}{5}z z=4+\frac{7}{2}y$$

Substitute -\frac{12}{5}+\frac{3}{5}z for y in the equation z=4+\frac{7}{2}y.

$$z=4+\frac{7}{2}\left(-\frac{12}{5}+\frac{3}{5}z\right)$$

Solve z=4+\frac{7}{2}\left(-\frac{12}{5}+\frac{3}{5}z\right) for z.

$$z=4$$

Substitute 4 for z in the equation y=-\frac{12}{5}+\frac{3}{5}z.

$$y=-\frac{12}{5}+\frac{3}{5}\times 4$$

Calculate y from y=-\frac{12}{5}+\frac{3}{5}\times 4.

$$y=0$$

Substitute 0 for y and 4 for z in the equation x=\frac{4}{3}y-\frac{2}{3}z+\frac{5}{3}.

$$x=\frac{4}{3}\times 0-\frac{2}{3}\times 4+\frac{5}{3}$$

Calculate x from x=\frac{4}{3}\times 0-\frac{2}{3}\times 4+\frac{5}{3}.

$$x=-1$$

The system is now solved.

$$x=-1 y=0 z=4$$