Solve {l}{2x+4y+z=77}{4x+3y+7z=114}{2x+y+3z=48}

$$\left. \begin{array} { l } { 2 x + 4 y + z = 77 } \\ { 4 x + 3 y + 7 z = 114 } \\ { 2 x + y + 3 z = 48 } \end{array} \right.$$

Solve for x, y, z

$$x=10$$

$$y=13$$

$$z=5$$

Short Steps Using Substitution

Solve 2x+4y+z=77 for z.

$$z=-2x-4y+77$$

Substitute -2x-4y+77 for z in the second and third equation.

$$4x+3y+7\left(-2x-4y+77\right)=114 2x+y+3\left(-2x-4y+77\right)=48$$

Solve these equations for y and x respectively.

$$y=17-\frac{2}{5}x x=-\frac{11}{4}y+\frac{183}{4}$$

Substitute 17-\frac{2}{5}x for y in the equation x=-\frac{11}{4}y+\frac{183}{4}.

$$x=-\frac{11}{4}\left(17-\frac{2}{5}x\right)+\frac{183}{4}$$

Solve x=-\frac{11}{4}\left(17-\frac{2}{5}x\right)+\frac{183}{4} for x.

$$x=10$$

Substitute 10 for x in the equation y=17-\frac{2}{5}x.

$$y=17-\frac{2}{5}\times 10$$

Calculate y from y=17-\frac{2}{5}\times 10.

$$y=13$$

Substitute 13 for y and 10 for x in the equation z=-2x-4y+77.

$$z=-2\times 10-4\times 13+77$$

Calculate z from z=-2\times 10-4\times 13+77.

$$z=5$$

The system is now solved.

$$x=10 y=13 z=5$$