Solve {l}{5x+2y-3z=0}{2z+y=x}{4x=6}

$$\left\{ \begin{array} { l } { 5 x + 2 y - 3 z = 0 } \\ { 2 z + y = x } \\ { 4 x = 6 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$y = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$z = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Short Steps Using Substitution

Reorder the equations.

$$2z+y=x 5x+2y-3z=0 4x=6$$

Solve 2z+y=x for x.

$$x=2z+y$$

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

$$5\left(2z+y\right)+2y-3z=0 4\left(2z+y\right)=6$$

Solve these equations for y and z respectively.

$$y=-z z=-\frac{1}{2}y+\frac{3}{4}$$

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

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

Solve z=-\frac{1}{2}\left(-1\right)z+\frac{3}{4} for z.

$$z=\frac{3}{2}$$

Substitute \frac{3}{2} for z in the equation y=-z.

$$y=-\frac{3}{2}$$

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

$$x=2\times \frac{3}{2}-\frac{3}{2}$$

Calculate x from x=2\times \frac{3}{2}-\frac{3}{2}.

$$x=\frac{3}{2}$$

The system is now solved.

$$x=\frac{3}{2} y=-\frac{3}{2} z=\frac{3}{2}$$