Certainly! Let's solve the equation step-by-step to find [tex]\( x \)[/tex]:
Given the equation:
[tex]\[ \frac{a - x}{b} - \frac{x + b}{a} = 0 \][/tex]
1. Combine the Fractions:
First, we need to express the two terms with a common denominator. The common denominator between [tex]\( b \)[/tex] and [tex]\( a \)[/tex] is [tex]\( ab \)[/tex].
[tex]\[
\frac{a - x}{b} = \frac{a(a - x)}{ab}
\][/tex]
[tex]\[
\frac{x + b}{a} = \frac{b(x + b)}{ab}
\][/tex]
2. Rewrite the Equation:
[tex]\[
\frac{a(a - x) - b(x + b)}{ab} = 0
\][/tex]
3. Eliminate the Denominator:
To clear the fraction, multiply both sides by [tex]\( ab \)[/tex]:
[tex]\[
a(a - x) - b(x + b) = 0
\][/tex]
4. Distribute the Terms:
Expand both terms:
[tex]\[
a^2 - ax - bx - b^2 = 0
\][/tex]
5. Combine Like Terms:
Combine the terms involving [tex]\( x \)[/tex]:
[tex]\[
a^2 - (a + b)x - b^2 = 0
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Move the constant terms to the other side of the equation:
[tex]\[
a^2 - b^2 = (a + b)x
\][/tex]
Then, divide both sides by [tex]\( (a + b) \)[/tex]:
[tex]\[
x = \frac{a^2 - b^2}{a + b}
\][/tex]
7. Simplify the Expression:
Notice that [tex]\( a^2 - b^2 \)[/tex] is a difference of squares, which can be factored:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Substitute this back into the equation:
[tex]\[
x = \frac{(a - b)(a + b)}{a + b}
\][/tex]
Provided [tex]\( a \neq -b \)[/tex], cancel out the [tex]\( (a + b) \)[/tex] terms:
[tex]\[
x = a - b
\][/tex]
Thus, the solution to the equation [tex]\(\frac{a-x}{b} - \frac{x+b}{a} = 0\)[/tex] is:
[tex]\[ x = a - b \][/tex]
