Solve {l}{2x=y-3}{x+z=2y+10}{2x+2y=z-7}

$$\left\{ \begin{array} { l } { 2 x = y - 3 } \\ { x + z = 2 y + 10 } \\ { 2 x + 2 y = z - 7 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=5$$

$$z=19$$

Short Steps Using Substitution

Solve 2x=y-3 for y.

$$y=2x+3$$

Substitute 2x+3 for y in the second and third equation.

$$x+z=2\left(2x+3\right)+10 2x+2\left(2x+3\right)=z-7$$

Solve these equations for x and z respectively.

$$x=-\frac{16}{3}+\frac{1}{3}z z=6x+13$$

Substitute -\frac{16}{3}+\frac{1}{3}z for x in the equation z=6x+13.

$$z=6\left(-\frac{16}{3}+\frac{1}{3}z\right)+13$$

Solve z=6\left(-\frac{16}{3}+\frac{1}{3}z\right)+13 for z.

$$z=19$$

Substitute 19 for z in the equation x=-\frac{16}{3}+\frac{1}{3}z.

$$x=-\frac{16}{3}+\frac{1}{3}\times 19$$

Calculate x from x=-\frac{16}{3}+\frac{1}{3}\times 19.

$$x=1$$

Substitute 1 for x in the equation y=2x+3.

$$y=2\times 1+3$$

Calculate y from y=2\times 1+3.

$$y=5$$

The system is now solved.

$$x=1 y=5 z=19$$