Solve x*(b+c)=(axb)+(axc)

$$x \times ( b + c ) = ( a x b ) + ( a x c )$$

Solve for a (complex solution)

$$\left\{\begin{matrix}\\a=1\text{, }&\text{unconditionally}\\a\in \mathrm{C}\text{, }&b=-c\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x by b+c.

$$xb+xc=axb+axc$$

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

$$axb+axc=xb+xc$$

Combine all terms containing a.

$$\left(xb+xc\right)a=xb+xc$$

The equation is in standard form.

$$\left(bx+cx\right)a=bx+cx$$

Divide both sides by xb+xc.

$$\frac{\left(bx+cx\right)a}{bx+cx}=\frac{x\left(b+c\right)}{bx+cx}$$

Dividing by xb+xc undoes the multiplication by xb+xc.

$$a=\frac{x\left(b+c\right)}{bx+cx}$$

Divide x\left(b+c\right) by xb+xc.

$$a=1$$

Solve for b (complex solution)

$$\left\{\begin{matrix}\\b=-c\text{, }&\text{unconditionally}\\b\in \mathrm{C}\text{, }&a=1\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x by b+c.

$$xb+xc=axb+axc$$

Subtract axb from both sides.

$$xb+xc-axb=axc$$

Subtract xc from both sides.

$$xb-axb=axc-xc$$

Combine all terms containing b.

$$\left(x-ax\right)b=axc-xc$$

The equation is in standard form.

$$\left(x-ax\right)b=acx-cx$$

Divide both sides by x-ax.

$$\frac{\left(x-ax\right)b}{x-ax}=\frac{cx\left(a-1\right)}{x-ax}$$

Dividing by x-ax undoes the multiplication by x-ax.

$$b=\frac{cx\left(a-1\right)}{x-ax}$$

Divide xc\left(-1+a\right) by x-ax.

$$b=-c$$

Solve for a

$$\left\{\begin{matrix}\\a=1\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&b=-c\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x by b+c.

$$xb+xc=axb+axc$$

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

$$axb+axc=xb+xc$$

Combine all terms containing a.

$$\left(xb+xc\right)a=xb+xc$$

The equation is in standard form.

$$\left(bx+cx\right)a=bx+cx$$

Divide both sides by xb+xc.

$$\frac{\left(bx+cx\right)a}{bx+cx}=\frac{x\left(b+c\right)}{bx+cx}$$

Dividing by xb+xc undoes the multiplication by xb+xc.

$$a=\frac{x\left(b+c\right)}{bx+cx}$$

Divide x\left(b+c\right) by xb+xc.

$$a=1$$

Solve for b

$$\left\{\begin{matrix}\\b=-c\text{, }&\text{unconditionally}\\b\in \mathrm{R}\text{, }&a=1\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x by b+c.

$$xb+xc=axb+axc$$

Subtract axb from both sides.

$$xb+xc-axb=axc$$

Subtract xc from both sides.

$$xb-axb=axc-xc$$

Combine all terms containing b.

$$\left(x-ax\right)b=axc-xc$$

The equation is in standard form.

$$\left(x-ax\right)b=acx-cx$$

Divide both sides by x-ax.

$$\frac{\left(x-ax\right)b}{x-ax}=\frac{cx\left(a-1\right)}{x-ax}$$

Dividing by x-ax undoes the multiplication by x-ax.

$$b=\frac{cx\left(a-1\right)}{x-ax}$$

Divide xc\left(-1+a\right) by x-ax.

$$b=-c$$