Solve {l}{8x+8y+6z=52}{6x+7y+2=16}{3x+9y+22=5}

$$\left\{ \begin{array} { l } { 8 x + 8 y + 6 z = 52 } \\ { 6 x + 7 y + 2 = 16 } \\ { 3 x + 9 y + 22 = 5 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{245}{33} = 7\frac{14}{33} \approx 7.424242424$$

$$y = -\frac{48}{11} = -4\frac{4}{11} \approx -4.363636364$$

$$z = \frac{454}{99} = 4\frac{58}{99} \approx 4.585858586$$

Short Steps Using Substitution

Solve 8x+8y+6z=52 for x.

$$x=\frac{13}{2}-y-\frac{3}{4}z$$

Substitute \frac{13}{2}-y-\frac{3}{4}z for x in the second and third equation.

$$6\left(\frac{13}{2}-y-\frac{3}{4}z\right)+7y+2=16 3\left(\frac{13}{2}-y-\frac{3}{4}z\right)+9y+22=5$$

Solve these equations for y and z respectively.

$$y=-25+\frac{9}{2}z z=\frac{146}{9}+\frac{8}{3}y$$

Substitute -25+\frac{9}{2}z for y in the equation z=\frac{146}{9}+\frac{8}{3}y.

$$z=\frac{146}{9}+\frac{8}{3}\left(-25+\frac{9}{2}z\right)$$

Solve z=\frac{146}{9}+\frac{8}{3}\left(-25+\frac{9}{2}z\right) for z.

$$z=\frac{454}{99}$$

Substitute \frac{454}{99} for z in the equation y=-25+\frac{9}{2}z.

$$y=-25+\frac{9}{2}\times \frac{454}{99}$$

Calculate y from y=-25+\frac{9}{2}\times \frac{454}{99}.

$$y=-\frac{48}{11}$$

Substitute -\frac{48}{11} for y and \frac{454}{99} for z in the equation x=\frac{13}{2}-y-\frac{3}{4}z.

$$x=\frac{13}{2}-\left(-\frac{48}{11}\right)-\frac{3}{4}\times \frac{454}{99}$$

Calculate x from x=\frac{13}{2}-\left(-\frac{48}{11}\right)-\frac{3}{4}\times \frac{454}{99}.

$$x=\frac{245}{33}$$

The system is now solved.

$$x=\frac{245}{33} y=-\frac{48}{11} z=\frac{454}{99}$$