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

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

Solve for x, y, z

$$x=\frac{2}{3}\approx 0.666666667$$

$$y = \frac{11}{3} = 3\frac{2}{3} \approx 3.666666667$$

$$z = \frac{10}{3} = 3\frac{1}{3} \approx 3.333333333$$

Short Steps Using Substitution

Solve x+y-z=1 for x.

$$x=-y+z+1$$

Substitute -y+z+1 for x in the second and third equation.

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

Solve these equations for y and z respectively.

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

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

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

Solve z=-4+2\left(2z-3\right) for z.

$$z=\frac{10}{3}$$

Substitute \frac{10}{3} for z in the equation y=2z-3.

$$y=2\times \frac{10}{3}-3$$

Calculate y from y=2\times \frac{10}{3}-3.

$$y=\frac{11}{3}$$

Substitute \frac{11}{3} for y and \frac{10}{3} for z in the equation x=-y+z+1.

$$x=-\frac{11}{3}+\frac{10}{3}+1$$

Calculate x from x=-\frac{11}{3}+\frac{10}{3}+1.

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

The system is now solved.

$$x=\frac{2}{3} y=\frac{11}{3} z=\frac{10}{3}$$