Solve {l}{2(x-z)=1}{3(y-z)=1}{(x+y-1)=1}

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

Solve for x, z, y

$$x = \frac{13}{12} = 1\frac{1}{12} \approx 1.083333333$$

$$y=\frac{11}{12}\approx 0.916666667$$

$$z=\frac{7}{12}\approx 0.583333333$$

Short Steps Using Substitution

Reorder the equations.

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

Solve x+y-1=1 for x.

$$x=-y+2$$

Substitute -y+2 for x in the equation 2\left(x-z\right)=1.

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

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

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

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

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

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

$$y=\frac{11}{12}$$

Substitute \frac{11}{12} for y in the equation z=y-\frac{1}{3}.

$$z=\frac{11}{12}-\frac{1}{3}$$

Calculate z from z=\frac{11}{12}-\frac{1}{3}.

$$z=\frac{7}{12}$$

Substitute \frac{11}{12} for y in the equation x=-y+2.

$$x=-\frac{11}{12}+2$$

Calculate x from x=-\frac{11}{12}+2.

$$x=\frac{13}{12}$$

The system is now solved.

$$x=\frac{13}{12} z=\frac{7}{12} y=\frac{11}{12}$$