Solve for x
x=\frac{z+1}{z-2}
z\neq 2
Solve for z
z=\frac{2x+1}{x-1}
x\neq 1
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\left(z-2\right)x=z+1
Combine all terms containing x.
\frac{\left(z-2\right)x}{z-2}=\frac{z+1}{z-2}
Divide both sides by z-2.
x=\frac{z+1}{z-2}
Dividing by z-2 undoes the multiplication by z-2.
xz-2x-z=1
Subtract z from both sides.
xz-z=1+2x
Add 2x to both sides.
\left(x-1\right)z=1+2x
Combine all terms containing z.
\left(x-1\right)z=2x+1
The equation is in standard form.
\frac{\left(x-1\right)z}{x-1}=\frac{2x+1}{x-1}
Divide both sides by x-1.
z=\frac{2x+1}{x-1}
Dividing by x-1 undoes the multiplication by x-1.
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