Solve a_{3}=1/2x_2^2x(y-3/2)

$$a _ { 3 } = \frac { 1 } { 2 } x _ { 2 } ^ { 2 } x ( y - \frac { 3 } { 2 } )$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{4a_{3}}{\left(2y-3\right)x_{2}^{2}}\text{, }&y\neq \frac{3}{2}\text{ and }x_{2}\neq 0\\x\in \mathrm{C}\text{, }&\left(x_{2}=0\text{ or }y=\frac{3}{2}\right)\text{ and }a_{3}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{2}x_{2}^{2}x by y-\frac{3}{2}.

$$a_{3}=\frac{1}{2}x_{2}^{2}xy-\frac{3}{4}x_{2}^{2}x$$

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

$$\frac{1}{2}x_{2}^{2}xy-\frac{3}{4}x_{2}^{2}x=a_{3}$$

Combine all terms containing x.

$$\left(\frac{1}{2}x_{2}^{2}y-\frac{3}{4}x_{2}^{2}\right)x=a_{3}$$

The equation is in standard form.

$$\left(\frac{yx_{2}^{2}}{2}-\frac{3x_{2}^{2}}{4}\right)x=a_{3}$$

Divide both sides by \frac{1}{2}x_{2}^{2}y-\frac{3}{4}x_{2}^{2}.

$$\frac{\left(\frac{yx_{2}^{2}}{2}-\frac{3x_{2}^{2}}{4}\right)x}{\frac{yx_{2}^{2}}{2}-\frac{3x_{2}^{2}}{4}}=\frac{a_{3}}{\frac{yx_{2}^{2}}{2}-\frac{3x_{2}^{2}}{4}}$$

Dividing by \frac{1}{2}x_{2}^{2}y-\frac{3}{4}x_{2}^{2} undoes the multiplication by \frac{1}{2}x_{2}^{2}y-\frac{3}{4}x_{2}^{2}.

$$x=\frac{a_{3}}{\frac{yx_{2}^{2}}{2}-\frac{3x_{2}^{2}}{4}}$$

Divide a_{3} by \frac{1}{2}x_{2}^{2}y-\frac{3}{4}x_{2}^{2}.

$$x=\frac{4a_{3}}{\left(2y-3\right)x_{2}^{2}}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{4a_{3}}{\left(2y-3\right)x_{2}^{2}}\text{, }&y\neq \frac{3}{2}\text{ and }x_{2}\neq 0\\x\in \mathrm{R}\text{, }&\left(x_{2}=0\text{ or }y=\frac{3}{2}\right)\text{ and }a_{3}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{2}x_{2}^{2}x by y-\frac{3}{2}.

$$a_{3}=\frac{1}{2}x_{2}^{2}xy-\frac{3}{4}x_{2}^{2}x$$

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

$$\frac{1}{2}x_{2}^{2}xy-\frac{3}{4}x_{2}^{2}x=a_{3}$$

Combine all terms containing x.

$$\left(\frac{1}{2}x_{2}^{2}y-\frac{3}{4}x_{2}^{2}\right)x=a_{3}$$

The equation is in standard form.

$$\left(\frac{yx_{2}^{2}}{2}-\frac{3x_{2}^{2}}{4}\right)x=a_{3}$$

Divide both sides by \frac{1}{2}x_{2}^{2}y-\frac{3}{4}x_{2}^{2}.

$$\frac{\left(\frac{yx_{2}^{2}}{2}-\frac{3x_{2}^{2}}{4}\right)x}{\frac{yx_{2}^{2}}{2}-\frac{3x_{2}^{2}}{4}}=\frac{a_{3}}{\frac{yx_{2}^{2}}{2}-\frac{3x_{2}^{2}}{4}}$$

Dividing by \frac{1}{2}x_{2}^{2}y-\frac{3}{4}x_{2}^{2} undoes the multiplication by \frac{1}{2}x_{2}^{2}y-\frac{3}{4}x_{2}^{2}.

$$x=\frac{a_{3}}{\frac{yx_{2}^{2}}{2}-\frac{3x_{2}^{2}}{4}}$$

Divide a_{3} by \frac{1}{2}x_{2}^{2}y-\frac{3}{4}x_{2}^{2}.

$$x=\frac{4a_{3}}{\left(2y-3\right)x_{2}^{2}}$$

Solve for a_3

$$a_{3}=\frac{x\left(2y-3\right)x_{2}^{2}}{4}$$