Solve {c}{x+y-2=5}{2x+2y-2z=7}{x-3y+5z=15}

$$\left. \begin{array} { c } { x + y - 2 = 5 } \\ { 2 x + 2 y - 2 z = 7 } \\ { x - 3 y + 5 z = 15 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{37}{8} = 4\frac{5}{8} = 4.625$$

$$y = \frac{19}{8} = 2\frac{3}{8} = 2.375$$

$$z = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

Short Steps Using Substitution

Solve x+y-2=5 for x.

$$x=-y+7$$

Substitute -y+7 for x in the second and third equation.

$$2\left(-y+7\right)+2y-2z=7 -y+7-3y+5z=15$$

Solve these equations for z and y respectively.

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

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

$$y=-2+\frac{5}{4}\times \frac{7}{2}$$

Calculate y from y=-2+\frac{5}{4}\times \frac{7}{2}.

$$y=\frac{19}{8}$$

Substitute \frac{19}{8} for y in the equation x=-y+7.

$$x=-\frac{19}{8}+7$$

Calculate x from x=-\frac{19}{8}+7.

$$x=\frac{37}{8}$$

The system is now solved.

$$x=\frac{37}{8} y=\frac{19}{8} z=\frac{7}{2}$$