Solve {c}{5x+6y+7z=18}{10x+12y+3z=25}{20x+17y+19z=56}

$$\left\{ \begin{array} { c } { 5 x + 6 y + 7 z = 18 } \\ { 10 x + 12 y + 3 z = 25 } \\ { 20 x + 17 y + 19 z = 56 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=1$$

$$z=1$$

Short Steps Using Substitution

Solve 5x+6y+7z=18 for x.

$$x=-\frac{6}{5}y-\frac{7}{5}z+\frac{18}{5}$$

Substitute -\frac{6}{5}y-\frac{7}{5}z+\frac{18}{5} for x in the second and third equation.

$$10\left(-\frac{6}{5}y-\frac{7}{5}z+\frac{18}{5}\right)+12y+3z=25 20\left(-\frac{6}{5}y-\frac{7}{5}z+\frac{18}{5}\right)+17y+19z=56$$

Solve these equations for z and y respectively.

$$z=1 y=\frac{16}{7}-\frac{9}{7}z$$

Substitute 1 for z in the equation y=\frac{16}{7}-\frac{9}{7}z.

$$y=\frac{16}{7}-\frac{9}{7}$$

Calculate y from y=\frac{16}{7}-\frac{9}{7}.

$$y=1$$

Substitute 1 for z and 1 for y in the equation x=-\frac{6}{5}y-\frac{7}{5}z+\frac{18}{5}.

$$x=-\frac{6}{5}-\frac{7}{5}+\frac{18}{5}$$

Calculate x from x=-\frac{6}{5}-\frac{7}{5}+\frac{18}{5}.

$$x=1$$

The system is now solved.

$$x=1 y=1 z=1$$