Solve {l}{x+2y+7z=17}{18x+7y+z=8}{x+6a+z=7}

$$\left\{ \begin{array} { l } { x + 2 y + 7 z = 17 } \\ { 18 x + 7 y + z = 8 } \\ { x + 6 a + z = 7 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{113-141a}{38}$$

$$y=\frac{375a-269}{38}$$

$$z=\frac{153-87a}{38}$$

Short Steps Using Substitution

Solve x+2y+7z=17 for x.

$$x=-2y-7z+17$$

Substitute -2y-7z+17 for x in the second and third equation.

$$18\left(-2y-7z+17\right)+7y+z=8 -2y-7z+17+6a+z=7$$

Solve these equations for y and z respectively.

$$y=-\frac{125}{29}z+\frac{298}{29} z=\frac{5}{3}-\frac{1}{3}y+a$$

Substitute -\frac{125}{29}z+\frac{298}{29} for y in the equation z=\frac{5}{3}-\frac{1}{3}y+a.

$$z=\frac{5}{3}-\frac{1}{3}\left(-\frac{125}{29}z+\frac{298}{29}\right)+a$$

Solve z=\frac{5}{3}-\frac{1}{3}\left(-\frac{125}{29}z+\frac{298}{29}\right)+a for z.

$$z=\frac{153}{38}-\frac{87}{38}a$$

Substitute \frac{153}{38}-\frac{87}{38}a for z in the equation y=-\frac{125}{29}z+\frac{298}{29}.

$$y=-\frac{125}{29}\left(\frac{153}{38}-\frac{87}{38}a\right)+\frac{298}{29}$$

Calculate y from y=-\frac{125}{29}\left(\frac{153}{38}-\frac{87}{38}a\right)+\frac{298}{29}.

$$y=-\frac{269}{38}+\frac{375}{38}a$$

Substitute -\frac{269}{38}+\frac{375}{38}a for y and \frac{153}{38}-\frac{87}{38}a for z in the equation x=-2y-7z+17.

$$x=-2\left(-\frac{269}{38}+\frac{375}{38}a\right)-7\left(\frac{153}{38}-\frac{87}{38}a\right)+17$$

Calculate x from x=-2\left(-\frac{269}{38}+\frac{375}{38}a\right)-7\left(\frac{153}{38}-\frac{87}{38}a\right)+17.

$$x=\frac{113}{38}-\frac{141}{38}a$$

The system is now solved.

$$x=\frac{113}{38}-\frac{141}{38}a y=-\frac{269}{38}+\frac{375}{38}a z=\frac{153}{38}-\frac{87}{38}a$$