Solve (a-b)(x-c)-(b-c)(x-a)=(c-a)(x-b)

$$( a - b ) ( x - c ) - ( b - c ) ( x - a ) = ( c - a ) ( x - b )$$

Solve for a (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply a-b by x-c.

$$ax-ac-bx+cb-\left(b-c\right)\left(x-a\right)=\left(c-a\right)\left(x-b\right)$$

Use the distributive property to multiply b-c by x-a.

$$ax-ac-bx+cb-\left(bx-ba-cx+ca\right)=\left(c-a\right)\left(x-b\right)$$

To find the opposite of bx-ba-cx+ca, find the opposite of each term.

$$ax-ac-bx+cb-bx+ba+cx-ca=\left(c-a\right)\left(x-b\right)$$

Combine -bx and -bx to get -2bx.

$$ax-ac-2bx+cb+ba+cx-ca=\left(c-a\right)\left(x-b\right)$$

Combine -ac and -ca to get -2ac.

$$ax-2ac-2bx+cb+ba+cx=\left(c-a\right)\left(x-b\right)$$

Use the distributive property to multiply c-a by x-b.

$$ax-2ac-2bx+cb+ba+cx=cx-cb-ax+ab$$

Add ax to both sides.

$$ax-2ac-2bx+cb+ba+cx+ax=cx-cb+ab$$

Combine ax and ax to get 2ax.

$$2ax-2ac-2bx+cb+ba+cx=cx-cb+ab$$

Subtract ab from both sides.

$$2ax-2ac-2bx+cb+ba+cx-ab=cx-cb$$

Combine ba and -ab to get 0.

$$2ax-2ac-2bx+cb+cx=cx-cb$$

Add 2bx to both sides.

$$2ax-2ac+cb+cx=cx-cb+2bx$$

Subtract cb from both sides.

$$2ax-2ac+cx=cx-cb+2bx-cb$$

Combine -cb and -cb to get -2cb.

$$2ax-2ac+cx=cx-2cb+2bx$$

Subtract cx from both sides.

$$2ax-2ac=cx-2cb+2bx-cx$$

Combine cx and -cx to get 0.

$$2ax-2ac=-2cb+2bx$$

Combine all terms containing a.

$$\left(2x-2c\right)a=-2cb+2bx$$

The equation is in standard form.

$$\left(2x-2c\right)a=2bx-2bc$$

Divide both sides by 2x-2c.

$$\frac{\left(2x-2c\right)a}{2x-2c}=\frac{2b\left(x-c\right)}{2x-2c}$$

Dividing by 2x-2c undoes the multiplication by 2x-2c.

$$a=\frac{2b\left(x-c\right)}{2x-2c}$$

Divide 2b\left(-c+x\right) by 2x-2c.

$$a=b$$

Solve for b (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply a-b by x-c.

$$ax-ac-bx+cb-\left(b-c\right)\left(x-a\right)=\left(c-a\right)\left(x-b\right)$$

Use the distributive property to multiply b-c by x-a.

$$ax-ac-bx+cb-\left(bx-ba-cx+ca\right)=\left(c-a\right)\left(x-b\right)$$

To find the opposite of bx-ba-cx+ca, find the opposite of each term.

$$ax-ac-bx+cb-bx+ba+cx-ca=\left(c-a\right)\left(x-b\right)$$

Combine -bx and -bx to get -2bx.

$$ax-ac-2bx+cb+ba+cx-ca=\left(c-a\right)\left(x-b\right)$$

Combine -ac and -ca to get -2ac.

$$ax-2ac-2bx+cb+ba+cx=\left(c-a\right)\left(x-b\right)$$

Use the distributive property to multiply c-a by x-b.

$$ax-2ac-2bx+cb+ba+cx=cx-cb-ax+ab$$

Add cb to both sides.

$$ax-2ac-2bx+cb+ba+cx+cb=cx-ax+ab$$

Combine cb and cb to get 2cb.

$$ax-2ac-2bx+2cb+ba+cx=cx-ax+ab$$

Subtract ab from both sides.

$$ax-2ac-2bx+2cb+ba+cx-ab=cx-ax$$

Combine ba and -ab to get 0.

$$ax-2ac-2bx+2cb+cx=cx-ax$$

Subtract ax from both sides.

$$-2ac-2bx+2cb+cx=cx-ax-ax$$

Combine -ax and -ax to get -2ax.

$$-2ac-2bx+2cb+cx=cx-2ax$$

Add 2ac to both sides.

$$-2bx+2cb+cx=cx-2ax+2ac$$

Subtract cx from both sides.

$$-2bx+2cb=cx-2ax+2ac-cx$$

Combine cx and -cx to get 0.

$$-2bx+2cb=-2ax+2ac$$

Combine all terms containing b.

$$\left(-2x+2c\right)b=-2ax+2ac$$

The equation is in standard form.

$$\left(2c-2x\right)b=2ac-2ax$$

Divide both sides by -2x+2c.

$$\frac{\left(2c-2x\right)b}{2c-2x}=\frac{2a\left(c-x\right)}{2c-2x}$$

Dividing by -2x+2c undoes the multiplication by -2x+2c.

$$b=\frac{2a\left(c-x\right)}{2c-2x}$$

Divide 2a\left(-x+c\right) by -2x+2c.

$$b=a$$

Solve for a

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

Steps for Solving Linear Equation

Use the distributive property to multiply a-b by x-c.

$$ax-ac-bx+cb-\left(b-c\right)\left(x-a\right)=\left(c-a\right)\left(x-b\right)$$

Use the distributive property to multiply b-c by x-a.

$$ax-ac-bx+cb-\left(bx-ba-cx+ca\right)=\left(c-a\right)\left(x-b\right)$$

To find the opposite of bx-ba-cx+ca, find the opposite of each term.

$$ax-ac-bx+cb-bx+ba+cx-ca=\left(c-a\right)\left(x-b\right)$$

Combine -bx and -bx to get -2bx.

$$ax-ac-2bx+cb+ba+cx-ca=\left(c-a\right)\left(x-b\right)$$

Combine -ac and -ca to get -2ac.

$$ax-2ac-2bx+cb+ba+cx=\left(c-a\right)\left(x-b\right)$$

Use the distributive property to multiply c-a by x-b.

$$ax-2ac-2bx+cb+ba+cx=cx-cb-ax+ab$$

Add ax to both sides.

$$ax-2ac-2bx+cb+ba+cx+ax=cx-cb+ab$$

Combine ax and ax to get 2ax.

$$2ax-2ac-2bx+cb+ba+cx=cx-cb+ab$$

Subtract ab from both sides.

$$2ax-2ac-2bx+cb+ba+cx-ab=cx-cb$$

Combine ba and -ab to get 0.

$$2ax-2ac-2bx+cb+cx=cx-cb$$

Add 2bx to both sides.

$$2ax-2ac+cb+cx=cx-cb+2bx$$

Subtract cb from both sides.

$$2ax-2ac+cx=cx-cb+2bx-cb$$

Combine -cb and -cb to get -2cb.

$$2ax-2ac+cx=cx-2cb+2bx$$

Subtract cx from both sides.

$$2ax-2ac=cx-2cb+2bx-cx$$

Combine cx and -cx to get 0.

$$2ax-2ac=-2cb+2bx$$

Combine all terms containing a.

$$\left(2x-2c\right)a=-2cb+2bx$$

The equation is in standard form.

$$\left(2x-2c\right)a=2bx-2bc$$

Divide both sides by 2x-2c.

$$\frac{\left(2x-2c\right)a}{2x-2c}=\frac{2b\left(x-c\right)}{2x-2c}$$

Dividing by 2x-2c undoes the multiplication by 2x-2c.

$$a=\frac{2b\left(x-c\right)}{2x-2c}$$

Divide 2b\left(-c+x\right) by 2x-2c.

$$a=b$$

Solve for b

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

Steps for Solving Linear Equation

Use the distributive property to multiply a-b by x-c.

$$ax-ac-bx+cb-\left(b-c\right)\left(x-a\right)=\left(c-a\right)\left(x-b\right)$$

Use the distributive property to multiply b-c by x-a.

$$ax-ac-bx+cb-\left(bx-ba-cx+ca\right)=\left(c-a\right)\left(x-b\right)$$

To find the opposite of bx-ba-cx+ca, find the opposite of each term.

$$ax-ac-bx+cb-bx+ba+cx-ca=\left(c-a\right)\left(x-b\right)$$

Combine -bx and -bx to get -2bx.

$$ax-ac-2bx+cb+ba+cx-ca=\left(c-a\right)\left(x-b\right)$$

Combine -ac and -ca to get -2ac.

$$ax-2ac-2bx+cb+ba+cx=\left(c-a\right)\left(x-b\right)$$

Use the distributive property to multiply c-a by x-b.

$$ax-2ac-2bx+cb+ba+cx=cx-cb-ax+ab$$

Add cb to both sides.

$$ax-2ac-2bx+cb+ba+cx+cb=cx-ax+ab$$

Combine cb and cb to get 2cb.

$$ax-2ac-2bx+2cb+ba+cx=cx-ax+ab$$

Subtract ab from both sides.

$$ax-2ac-2bx+2cb+ba+cx-ab=cx-ax$$

Combine ba and -ab to get 0.

$$ax-2ac-2bx+2cb+cx=cx-ax$$

Subtract ax from both sides.

$$-2ac-2bx+2cb+cx=cx-ax-ax$$

Combine -ax and -ax to get -2ax.

$$-2ac-2bx+2cb+cx=cx-2ax$$

Add 2ac to both sides.

$$-2bx+2cb+cx=cx-2ax+2ac$$

Subtract cx from both sides.

$$-2bx+2cb=cx-2ax+2ac-cx$$

Combine cx and -cx to get 0.

$$-2bx+2cb=-2ax+2ac$$

Combine all terms containing b.

$$\left(-2x+2c\right)b=-2ax+2ac$$

The equation is in standard form.

$$\left(2c-2x\right)b=2ac-2ax$$

Divide both sides by -2x+2c.

$$\frac{\left(2c-2x\right)b}{2c-2x}=\frac{2a\left(c-x\right)}{2c-2x}$$

Dividing by -2x+2c undoes the multiplication by -2x+2c.

$$b=\frac{2a\left(c-x\right)}{2c-2x}$$

Divide 2a\left(-x+c\right) by -2x+2c.

$$b=a$$