Solve {l}{8x+13y+13z=154}{13x+8y+13z=154}{43x+36y+87z=846}

$$\left. \begin{array} { l } { 8 x + 13 y + 13 z = 154 } \\ { 13 x + 8 y + 13 z = 154 } \\ { 43 x + 36 y + 87 z = 846 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=3$$

$$z=7$$

Short Steps Using Substitution

Solve 8x+13y+13z=154 for x.

$$x=-\frac{13}{8}y-\frac{13}{8}z+\frac{77}{4}$$

Substitute -\frac{13}{8}y-\frac{13}{8}z+\frac{77}{4} for x in the second and third equation.

$$13\left(-\frac{13}{8}y-\frac{13}{8}z+\frac{77}{4}\right)+8y+13z=154 43\left(-\frac{13}{8}y-\frac{13}{8}z+\frac{77}{4}\right)+36y+87z=846$$

Solve these equations for y and z respectively.

$$y=\frac{22}{3}-\frac{13}{21}z z=\frac{146}{137}+\frac{271}{137}y$$

Substitute \frac{22}{3}-\frac{13}{21}z for y in the equation z=\frac{146}{137}+\frac{271}{137}y.

$$z=\frac{146}{137}+\frac{271}{137}\left(\frac{22}{3}-\frac{13}{21}z\right)$$

Solve z=\frac{146}{137}+\frac{271}{137}\left(\frac{22}{3}-\frac{13}{21}z\right) for z.

$$z=7$$

Substitute 7 for z in the equation y=\frac{22}{3}-\frac{13}{21}z.

$$y=\frac{22}{3}-\frac{13}{21}\times 7$$

Calculate y from y=\frac{22}{3}-\frac{13}{21}\times 7.

$$y=3$$

Substitute 3 for y and 7 for z in the equation x=-\frac{13}{8}y-\frac{13}{8}z+\frac{77}{4}.

$$x=-\frac{13}{8}\times 3-\frac{13}{8}\times 7+\frac{77}{4}$$

Calculate x from x=-\frac{13}{8}\times 3-\frac{13}{8}\times 7+\frac{77}{4}.

$$x=3$$

The system is now solved.

$$x=3 y=3 z=7$$