Solve {l}{5x+10y+15z=230}{3x-4y+5z=94}{7y+8y+9z=190}

$$\left. \begin{array} { l } { 5 x + 10 y + 15 z = 230 } \\ { 3 x - 4 y + 5 z = 94 } \\ { 7 y + 8 y + 9 z = 190 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{806}{15} = -53\frac{11}{15} \approx -53.733333333$$

$$y = -\frac{182}{15} = -12\frac{2}{15} \approx -12.133333333$$

$$z = \frac{124}{3} = 41\frac{1}{3} \approx 41.333333333$$

Short Steps Using Substitution

Solve 5x+10y+15z=230 for x.

$$x=46-2y-3z$$

Substitute 46-2y-3z for x in the equation 3x-4y+5z=94.

$$3\left(46-2y-3z\right)-4y+5z=94$$

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

$$y=\frac{22}{5}-\frac{2}{5}z z=\frac{190}{9}-\frac{5}{3}y$$

Substitute \frac{22}{5}-\frac{2}{5}z for y in the equation z=\frac{190}{9}-\frac{5}{3}y.

$$z=\frac{190}{9}-\frac{5}{3}\left(\frac{22}{5}-\frac{2}{5}z\right)$$

Solve z=\frac{190}{9}-\frac{5}{3}\left(\frac{22}{5}-\frac{2}{5}z\right) for z.

$$z=\frac{124}{3}$$

Substitute \frac{124}{3} for z in the equation y=\frac{22}{5}-\frac{2}{5}z.

$$y=\frac{22}{5}-\frac{2}{5}\times \frac{124}{3}$$

Calculate y from y=\frac{22}{5}-\frac{2}{5}\times \frac{124}{3}.

$$y=-\frac{182}{15}$$

Substitute -\frac{182}{15} for y and \frac{124}{3} for z in the equation x=46-2y-3z.

$$x=46-2\left(-\frac{182}{15}\right)-3\times \frac{124}{3}$$

Calculate x from x=46-2\left(-\frac{182}{15}\right)-3\times \frac{124}{3}.

$$x=-\frac{806}{15}$$

The system is now solved.

$$x=-\frac{806}{15} y=-\frac{182}{15} z=\frac{124}{3}$$