Solve {l}{7x-4y+3z=28}{3x+3y-z=19}{3x+2y+z=16}

$$\left. \begin{array} { l } { 7 x - 4 y + 3 z = 28 } \\ { 3 x + 3 y - z = 19 } \\ { 3 x + 2 y + z = 16 } \end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y=1$$

$$z=-1$$

Short Steps Using Substitution

Reorder the equations.

$$3x+3y-z=19 7x-4y+3z=28 3x+2y+z=16$$

Solve 3x+3y-z=19 for z.

$$z=3x+3y-19$$

Substitute 3x+3y-19 for z in the second and third equation.

$$7x-4y+3\left(3x+3y-19\right)=28 3x+2y+3x+3y-19=16$$

Solve these equations for y and x respectively.

$$y=-\frac{16}{5}x+17 x=\frac{35}{6}-\frac{5}{6}y$$

Substitute -\frac{16}{5}x+17 for y in the equation x=\frac{35}{6}-\frac{5}{6}y.

$$x=\frac{35}{6}-\frac{5}{6}\left(-\frac{16}{5}x+17\right)$$

Solve x=\frac{35}{6}-\frac{5}{6}\left(-\frac{16}{5}x+17\right) for x.

$$x=5$$

Substitute 5 for x in the equation y=-\frac{16}{5}x+17.

$$y=-\frac{16}{5}\times 5+17$$

Calculate y from y=-\frac{16}{5}\times 5+17.

$$y=1$$

Substitute 1 for y and 5 for x in the equation z=3x+3y-19.

$$z=3\times 5+3\times 1-19$$

Calculate z from z=3\times 5+3\times 1-19.

$$z=-1$$

The system is now solved.

$$x=5 y=1 z=-1$$