Solve for x (x-a)/b=2-(x-b)/a

$\frac{x - a}{b} = 2 - \frac{x - b}{a}$

Step 1

Multiply both sides by $b$ .

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

Step 2

Simplify.

Simplify the left side.

Simplify $\frac{x - a}{b} b$ .

Cancel the common factor of $b$ .

Cancel the common factor.

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

Rewrite the expression.

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

Reorder $x$ and $- a$ .

$- a + x = (2 - \frac{x - b}{a}) b$

Simplify the right side.

Simplify $(2 - \frac{x - b}{a}) b$ .

To write $2$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$- a + x = (\frac{2 a}{a} - \frac{x - b}{a}) b$

Combine the numerators over the common denominator.

$- a + x = \frac{2 a - (x - b)}{a} b$

Simplify the numerator.

Apply the distributive property.

$- a + x = \frac{2 a - x - - b}{a} b$

Multiply $- - b$ .

Multiply $- 1$ by $- 1$ .

$- a + x = \frac{2 a - x + 1 b}{a} b$

Multiply $b$ by $1$ .

$- a + x = \frac{2 a - x + b}{a} b$

Combine fractions.

Combine $\frac{2 a - x + b}{a}$ and $b$ .

$- a + x = \frac{(2 a - x + b) b}{a}$

Reorder factors in $\frac{(2 a - x + b) b}{a}$ .

$- a + x = \frac{b (2 a - x + b)}{a}$

Step 3

Solve for $x$ .

Multiply both sides by $a$ .

$(- a + x) a = \frac{b (2 a - x + b)}{a} a$

Simplify.

Simplify the left side.

Simplify $(- a + x) a$ .

Apply the distributive property.

$- a \cdot a + x a = \frac{b (2 a - x + b)}{a} a$

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$- (a \cdot a) + x a = \frac{b (2 a - x + b)}{a} a$

Multiply $a$ by $a$ .

$- a^{2} + x a = \frac{b (2 a - x + b)}{a} a$

Reorder $x$ and $a$ .

$- a^{2} + a x = \frac{b (2 a - x + b)}{a} a$

Reorder $- a^{2}$ and $a x$ .

$a x - a^{2} = \frac{b (2 a - x + b)}{a} a$

Simplify the right side.

Simplify $\frac{b (2 a - x + b)}{a} a$ .

Cancel the common factor of $a$ .

Cancel the common factor.

$a x - a^{2} = \frac{b (2 a - x + b)}{a} a$

Rewrite the expression.

$a x - a^{2} = b (2 a - x + b)$

Apply the distributive property.

$a x - a^{2} = b (2 a) + b (- x) + b \cdot b$

Simplify.

Rewrite using the commutative property of multiplication.

$a x - a^{2} = 2 b a + b (- x) + b \cdot b$

Rewrite using the commutative property of multiplication.

$a x - a^{2} = 2 b a - b x + b \cdot b$

Multiply $b$ by $b$ .

$a x - a^{2} = 2 b a - b x + b^{2}$

Move $b$ .

$a x - a^{2} = 2 a b - b x + b^{2}$

Solve for $x$ .

Add $b x$ to both sides of the equation.

$a x - a^{2} + b x = 2 a b + b^{2}$

Add $a^{2}$ to both sides of the equation.

$a x + b x = 2 a b + b^{2} + a^{2}$

Factor $x$ out of $a x + b x$ .

Factor $x$ out of $a x$ .

$x a + b x = 2 a b + b^{2} + a^{2}$

Factor $x$ out of $b x$ .

$x a + x b = 2 a b + b^{2} + a^{2}$

Factor $x$ out of $x a + x b$ .

$x (a + b) = 2 a b + b^{2} + a^{2}$

Divide each term in $x (a + b) = 2 a b + b^{2} + a^{2}$ by $a + b$ and simplify.

Divide each term in $x (a + b) = 2 a b + b^{2} + a^{2}$ by $a + b$ .

$\frac{x (a + b)}{a + b} = \frac{2 a b}{a + b} + \frac{b^{2}}{a + b} + \frac{a^{2}}{a + b}$

Simplify the left side.

Cancel the common factor of $a + b$ .

Cancel the common factor.

$\frac{x (a + b)}{a + b} = \frac{2 a b}{a + b} + \frac{b^{2}}{a + b} + \frac{a^{2}}{a + b}$

Divide $x$ by $1$ .

$x = \frac{2 a b}{a + b} + \frac{b^{2}}{a + b} + \frac{a^{2}}{a + b}$

Simplify the right side.

Combine into one fraction.

Combine the numerators over the common denominator.

$x = \frac{2 a b + b^{2}}{a + b} + \frac{a^{2}}{a + b}$

Combine the numerators over the common denominator.

$x = \frac{2 a b + b^{2} + a^{2}}{a + b}$

Factor using the perfect square rule.

Rearrange terms.

$x = \frac{b^{2} + 2 a b + a^{2}}{a + b}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot b \cdot a$

Rewrite the polynomial.

$x = \frac{b^{2} + 2 \cdot b \cdot a + a^{2}}{a + b}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = b$ and $b = a$ .

$x = \frac{{(b + a)}^{2}}{a + b}$

Cancel the common factor of ${(b + a)}^{2}$ and $a + b$ .

Reorder terms.

$x = \frac{{(a + b)}^{2}}{a + b}$

Factor $a + b$ out of ${(a + b)}^{2}$ .

$x = \frac{(a + b) (a + b)}{a + b}$

Cancel the common factors.

Multiply by $1$ .

$x = \frac{(a + b) (a + b)}{(a + b) \cdot 1}$

Cancel the common factor.

$x = \frac{(a + b) (a + b)}{(a + b) \cdot 1}$

Rewrite the expression.

$x = \frac{a + b}{1}$

Divide $a + b$ by $1$ .

$x = a + b$

Solve for x (x-a)/b=2-(x-b)/a

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