Solve {l}{x+y+z=9}{2x+5y+7z=52}{2x+y-z=0}

$$\left. \begin{array} { l } { x + y + z = 9 } \\ { 2 x + 5 y + 7 z = 52 } \\ { 2 x + y - z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=3$$

$$z=5$$

Short Steps Using Substitution

Solve x+y+z=9 for x.

$$x=-y-z+9$$

Substitute -y-z+9 for x in the second and third equation.

$$2\left(-y-z+9\right)+5y+7z=52 2\left(-y-z+9\right)+y-z=0$$

Solve these equations for y and z respectively.

$$y=-\frac{5}{3}z+\frac{34}{3} z=-\frac{1}{3}y+6$$

Substitute -\frac{5}{3}z+\frac{34}{3} for y in the equation z=-\frac{1}{3}y+6.

$$z=-\frac{1}{3}\left(-\frac{5}{3}z+\frac{34}{3}\right)+6$$

Solve z=-\frac{1}{3}\left(-\frac{5}{3}z+\frac{34}{3}\right)+6 for z.

$$z=5$$

Substitute 5 for z in the equation y=-\frac{5}{3}z+\frac{34}{3}.

$$y=-\frac{5}{3}\times 5+\frac{34}{3}$$

Calculate y from y=-\frac{5}{3}\times 5+\frac{34}{3}.

$$y=3$$

Substitute 3 for y and 5 for z in the equation x=-y-z+9.

$$x=-3-5+9$$

Calculate x from x=-3-5+9.

$$x=1$$

The system is now solved.

$$x=1 y=3 z=5$$