Solve {l}{3x+4z=7}{2x+3y+z=9}{5x-9y+7z=8}

$$\left. \begin{array} { l } { 3 x + 4 z = 7 } \\ { 2 x + 3 y + z = 9 } \\ { 5 x - 9 y + 7 z = 8 } \end{array} \right.$$

Solve for x, z, y

$$x=5$$

$$y=\frac{1}{3}\approx 0.333333333$$

$$z=-2$$

Short Steps Using Substitution

Reorder the equations.

$$2x+3y+z=9 3x+4z=7 5x-9y+7z=8$$

Solve 2x+3y+z=9 for z.

$$z=-2x-3y+9$$

Substitute -2x-3y+9 for z in the second and third equation.

$$3x+4\left(-2x-3y+9\right)=7 5x-9y+7\left(-2x-3y+9\right)=8$$

Solve these equations for x and y respectively.

$$x=-\frac{12}{5}y+\frac{29}{5} y=-\frac{3}{10}x+\frac{11}{6}$$

Substitute -\frac{12}{5}y+\frac{29}{5} for x in the equation y=-\frac{3}{10}x+\frac{11}{6}.

$$y=-\frac{3}{10}\left(-\frac{12}{5}y+\frac{29}{5}\right)+\frac{11}{6}$$

Solve y=-\frac{3}{10}\left(-\frac{12}{5}y+\frac{29}{5}\right)+\frac{11}{6} for y.

$$y=\frac{1}{3}$$

Substitute \frac{1}{3} for y in the equation x=-\frac{12}{5}y+\frac{29}{5}.

$$x=-\frac{12}{5}\times \frac{1}{3}+\frac{29}{5}$$

Calculate x from x=-\frac{12}{5}\times \frac{1}{3}+\frac{29}{5}.

$$x=5$$

Substitute 5 for x and \frac{1}{3} for y in the equation z=-2x-3y+9.

$$z=-2\times 5-3\times \frac{1}{3}+9$$

Calculate z from z=-2\times 5-3\times \frac{1}{3}+9.

$$z=-2$$

The system is now solved.

$$x=5 z=-2 y=\frac{1}{3}$$