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

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

Solve for x, y, z

$$x=2$$

$$y=1$$

$$z=3$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for y and z respectively.

$$y=\frac{11}{7}z-\frac{26}{7} z=y+2$$

Substitute \frac{11}{7}z-\frac{26}{7} for y in the equation z=y+2.

$$z=\frac{11}{7}z-\frac{26}{7}+2$$

Solve z=\frac{11}{7}z-\frac{26}{7}+2 for z.

$$z=3$$

Substitute 3 for z in the equation y=\frac{11}{7}z-\frac{26}{7}.

$$y=\frac{11}{7}\times 3-\frac{26}{7}$$

Calculate y from y=\frac{11}{7}\times 3-\frac{26}{7}.

$$y=1$$

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

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

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

$$x=2$$

The system is now solved.

$$x=2 y=1 z=3$$