Solve {r}{3x-4y+z=3}{2x+2y=8}{5y+8z=14}

$$\left. \begin{array} { r } { 3 x - 4 y + z = 3 } \\ { 2 x + 2 y = 8 } \\ { 5 y + 8 z = 14 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{158}{61} = 2\frac{36}{61} \approx 2.590163934$$

$$y = \frac{86}{61} = 1\frac{25}{61} \approx 1.409836066$$

$$z=\frac{53}{61}\approx 0.868852459$$

Short Steps Using Substitution

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

$$z=-3x+4y+3$$

Substitute -3x+4y+3 for z in the equation 5y+8z=14.

$$5y+8\left(-3x+4y+3\right)=14$$

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

$$y=4-x x=\frac{5}{12}+\frac{37}{24}y$$

Substitute 4-x for y in the equation x=\frac{5}{12}+\frac{37}{24}y.

$$x=\frac{5}{12}+\frac{37}{24}\left(4-x\right)$$

Solve x=\frac{5}{12}+\frac{37}{24}\left(4-x\right) for x.

$$x=\frac{158}{61}$$

Substitute \frac{158}{61} for x in the equation y=4-x.

$$y=4-\frac{158}{61}$$

Calculate y from y=4-\frac{158}{61}.

$$y=\frac{86}{61}$$

Substitute \frac{86}{61} for y and \frac{158}{61} for x in the equation z=-3x+4y+3.

$$z=-3\times \frac{158}{61}+4\times \frac{86}{61}+3$$

Calculate z from z=-3\times \frac{158}{61}+4\times \frac{86}{61}+3.

$$z=\frac{53}{61}$$

The system is now solved.

$$x=\frac{158}{61} y=\frac{86}{61} z=\frac{53}{61}$$