Solve X=a+b-c^2/2-a*b

$$X = a + \frac { b - c ^ { 2 } } { 2 } - a \cdot b$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{b-2X-c^{2}}{2\left(1-b\right)}\text{, }&b\neq 1\\a\in \mathrm{R}\text{, }&X=\frac{1-c^{2}}{2}\text{ and }b=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Divide each term of b-c^{2} by 2 to get \frac{1}{2}b-\frac{1}{2}c^{2}.

$$X=a+\frac{1}{2}b-\frac{1}{2}c^{2}-ab$$

Swap sides so that all variable terms are on the left hand side.

$$a+\frac{1}{2}b-\frac{1}{2}c^{2}-ab=X$$

Subtract \frac{1}{2}b from both sides.

$$a-\frac{1}{2}c^{2}-ab=X-\frac{1}{2}b$$

Add \frac{1}{2}c^{2} to both sides.

$$a-ab=X-\frac{1}{2}b+\frac{1}{2}c^{2}$$

Combine all terms containing a.

$$\left(1-b\right)a=X-\frac{1}{2}b+\frac{1}{2}c^{2}$$

The equation is in standard form.

$$\left(1-b\right)a=\frac{c^{2}}{2}-\frac{b}{2}+X$$

Divide both sides by 1-b.

$$\frac{\left(1-b\right)a}{1-b}=\frac{\frac{c^{2}}{2}-\frac{b}{2}+X}{1-b}$$

Dividing by 1-b undoes the multiplication by 1-b.

$$a=\frac{\frac{c^{2}}{2}-\frac{b}{2}+X}{1-b}$$

Divide X-\frac{b}{2}+\frac{c^{2}}{2} by 1-b.

$$a=\frac{2X-b+c^{2}}{2\left(1-b\right)}$$

Solve for X

$$X=-ab-\frac{c^{2}}{2}+\frac{b}{2}+a$$

Solution Steps

Divide each term of b-c^{2} by 2 to get \frac{1}{2}b-\frac{1}{2}c^{2}.

$$X=a+\frac{1}{2}b-\frac{1}{2}c^{2}-ab$$