Solve {l}{x+2y+x=4}{3x-4y-z=2}{5x+3y+5z=-1}

$$\left. \begin{array} { l } { x + 2 y + x = 4 } \\ { 3 x - 4 y - z = 2 } \\ { 5 x + 3 y + 5 z = - 1 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{43}{37} = 1\frac{6}{37} \approx 1.162162162$$

$$y=\frac{31}{37}\approx 0.837837838$$

$$z = -\frac{69}{37} = -1\frac{32}{37} \approx -1.864864865$$

Short Steps Using Substitution

Reorder the equations.

$$3x-4y-z=2 x+2y+x=4 5x+3y+5z=-1$$

Solve 3x-4y-z=2 for z.

$$z=3x-4y-2$$

Substitute 3x-4y-2 for z in the equation 5x+3y+5z=-1.

$$5x+3y+5\left(3x-4y-2\right)=-1$$

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

$$y=2-x x=\frac{17}{20}y+\frac{9}{20}$$

Substitute 2-x for y in the equation x=\frac{17}{20}y+\frac{9}{20}.

$$x=\frac{17}{20}\left(2-x\right)+\frac{9}{20}$$

Solve x=\frac{17}{20}\left(2-x\right)+\frac{9}{20} for x.

$$x=\frac{43}{37}$$

Substitute \frac{43}{37} for x in the equation y=2-x.

$$y=2-\frac{43}{37}$$

Calculate y from y=2-\frac{43}{37}.

$$y=\frac{31}{37}$$

Substitute \frac{31}{37} for y and \frac{43}{37} for x in the equation z=3x-4y-2.

$$z=3\times \frac{43}{37}-4\times \frac{31}{37}-2$$

Calculate z from z=3\times \frac{43}{37}-4\times \frac{31}{37}-2.

$$z=-\frac{69}{37}$$

The system is now solved.

$$x=\frac{43}{37} y=\frac{31}{37} z=-\frac{69}{37}$$