Solve {l}{x+2y=5}{3y-z=1}{2x-y+3z=41}

$$\left. \begin{array} { l } { x + 2 y = 5 } \\ { 3 y - z = 1 } \\ { 2 x - y + 3 z = 41 } \end{array} \right.$$

Solve for x, y, z

$$x=-12$$

$$y = \frac{17}{2} = 8\frac{1}{2} = 8.5$$

$$z = \frac{49}{2} = 24\frac{1}{2} = 24.5$$

Short Steps Using Substitution

Solve x+2y=5 for x.

$$x=-2y+5$$

Substitute -2y+5 for x in the equation 2x-y+3z=41.

$$2\left(-2y+5\right)-y+3z=41$$

Solve the second equation for y and the third equation for z.

$$y=\frac{1}{3}+\frac{1}{3}z z=\frac{5}{3}y+\frac{31}{3}$$

Substitute \frac{1}{3}+\frac{1}{3}z for y in the equation z=\frac{5}{3}y+\frac{31}{3}.

$$z=\frac{5}{3}\left(\frac{1}{3}+\frac{1}{3}z\right)+\frac{31}{3}$$

Solve z=\frac{5}{3}\left(\frac{1}{3}+\frac{1}{3}z\right)+\frac{31}{3} for z.

$$z=\frac{49}{2}$$

Substitute \frac{49}{2} for z in the equation y=\frac{1}{3}+\frac{1}{3}z.

$$y=\frac{1}{3}+\frac{1}{3}\times \frac{49}{2}$$

Calculate y from y=\frac{1}{3}+\frac{1}{3}\times \frac{49}{2}.

$$y=\frac{17}{2}$$

Substitute \frac{17}{2} for y in the equation x=-2y+5.

$$x=-2\times \frac{17}{2}+5$$

Calculate x from x=-2\times \frac{17}{2}+5.

$$x=-12$$

The system is now solved.

$$x=-12 y=\frac{17}{2} z=\frac{49}{2}$$