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

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

Solve for x, z, y

$$x = \frac{103}{11} = 9\frac{4}{11} \approx 9.363636364$$

$$y = -\frac{43}{11} = -3\frac{10}{11} \approx -3.909090909$$

$$z = -\frac{58}{11} = -5\frac{3}{11} \approx -5.272727273$$

Short Steps Using Substitution

Reorder the equations.

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

Solve x-y+z=8 for x.

$$x=y-z+8$$

Substitute y-z+8 for x in the second and third equation.

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

Solve these equations for z and y respectively.

$$z=\frac{9}{2}+\frac{5}{2}y y=-\frac{17}{3}-\frac{1}{3}z$$

Substitute \frac{9}{2}+\frac{5}{2}y for z in the equation y=-\frac{17}{3}-\frac{1}{3}z.

$$y=-\frac{17}{3}-\frac{1}{3}\left(\frac{9}{2}+\frac{5}{2}y\right)$$

Solve y=-\frac{17}{3}-\frac{1}{3}\left(\frac{9}{2}+\frac{5}{2}y\right) for y.

$$y=-\frac{43}{11}$$

Substitute -\frac{43}{11} for y in the equation z=\frac{9}{2}+\frac{5}{2}y.

$$z=\frac{9}{2}+\frac{5}{2}\left(-\frac{43}{11}\right)$$

Calculate z from z=\frac{9}{2}+\frac{5}{2}\left(-\frac{43}{11}\right).

$$z=-\frac{58}{11}$$

Substitute -\frac{58}{11} for z and -\frac{43}{11} for y in the equation x=y-z+8.

$$x=-\frac{43}{11}-\left(-\frac{58}{11}\right)+8$$

Calculate x from x=-\frac{43}{11}-\left(-\frac{58}{11}\right)+8.

$$x=\frac{103}{11}$$

The system is now solved.

$$x=\frac{103}{11} z=-\frac{58}{11} y=-\frac{43}{11}$$