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

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

Solve for x, y, z

$$x = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

$$y=-\frac{1}{3}\approx -0.333333333$$

$$z=1$$

Short Steps Using Substitution

Reorder the equations.

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

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

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

Substitute 2y-4z+3 for x in the second and third equation.

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

Solve these equations for z and y respectively.

$$z=1 y=z-\frac{4}{3}$$

Substitute 1 for z in the equation y=z-\frac{4}{3}.

$$y=1-\frac{4}{3}$$

Calculate y from y=1-\frac{4}{3}.

$$y=-\frac{1}{3}$$

Substitute 1 for z and -\frac{1}{3} for y in the equation x=2y-4z+3.

$$x=2\left(-\frac{1}{3}\right)-4+3$$

Calculate x from x=2\left(-\frac{1}{3}\right)-4+3.

$$x=-\frac{5}{3}$$

The system is now solved.

$$x=-\frac{5}{3} y=-\frac{1}{3} z=1$$