Solve {l}{x+y-2y=-3}{5x-2y+7z=22}{2x-5y+4z=4}

$$\left. \begin{array} { l } { x + y - 2 y = - 3 } \\ { 5 x - 2 y + 7 z = 22 } \\ { 2 x - 5 y + 4 z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{7}{11}\approx -0.636363636$$

$$y = \frac{26}{11} = 2\frac{4}{11} \approx 2.363636364$$

$$z = \frac{47}{11} = 4\frac{3}{11} \approx 4.272727273$$

Short Steps Using Substitution

Solve x+y-2y=-3 for x.

$$x=y-3$$

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

$$5\left(y-3\right)-2y+7z=22 2\left(y-3\right)-5y+4z=4$$

Solve these equations for y and z respectively.

$$y=\frac{37}{3}-\frac{7}{3}z z=\frac{3}{4}y+\frac{5}{2}$$

Substitute \frac{37}{3}-\frac{7}{3}z for y in the equation z=\frac{3}{4}y+\frac{5}{2}.

$$z=\frac{3}{4}\left(\frac{37}{3}-\frac{7}{3}z\right)+\frac{5}{2}$$

Solve z=\frac{3}{4}\left(\frac{37}{3}-\frac{7}{3}z\right)+\frac{5}{2} for z.

$$z=\frac{47}{11}$$

Substitute \frac{47}{11} for z in the equation y=\frac{37}{3}-\frac{7}{3}z.

$$y=\frac{37}{3}-\frac{7}{3}\times \frac{47}{11}$$

Calculate y from y=\frac{37}{3}-\frac{7}{3}\times \frac{47}{11}.

$$y=\frac{26}{11}$$

Substitute \frac{26}{11} for y in the equation x=y-3.

$$x=\frac{26}{11}-3$$

Calculate x from x=\frac{26}{11}-3.

$$x=-\frac{7}{11}$$

The system is now solved.

$$x=-\frac{7}{11} y=\frac{26}{11} z=\frac{47}{11}$$