Answer:
If solving for x, [tex]x = ab[/tex]
If solving for any solutions, [tex]x = ab \text{ or } a = \pm b[/tex]
Step-by-step explanation:
[tex]\frac{ax}b - \frac{bx}a = a^2 - b^2[/tex]
bring the fractions on the left to a single fraction with a common denominator by multiplying each of the fractions by the other's denominator.
[tex]\frac aa \times \frac{ax}b - \frac bb\times\frac{bx}a = a^2 - b^2\\\frac{a^2x - b^2x}{ab} = a^2 - b^2 \text{ // Factor }x \text{ on the left}\\\\\frac{a^2 - b^2}{ab}x = a^2-b^2 \text{ //}-(a^2 - b^2)\\\\\frac{a^2 - b^2}{ab}x-(a^2-b^2)=0\text{ // Factor }a^2 -b^2\text{ on the left}\\\\(a^2 - b^2)(\frac{x}{ab} - 1)=0[/tex]
Therefore,
[tex]a^2 -b^2 = 0 \to a = \pm b\\ \text{or}\\\frac x{ab} - 1 = 0\\\to \frac x{ab} = 1\\\to x = ab[/tex]
Answer:
x = ab
Step-by-step explanation:
Given equation:
[tex]\dfrac{ax}{b}-\dfrac{bx}{a}=a^2-b^2[/tex]
To solve the given equation for x, begin by finding a common denominator for the fractions on the left side, which is ab.
Multiply the numerator and denominator of the first fraction by 'a'. Similarly, multiply the numerator and denominator of the second fraction by 'b' so that both fractions have the common denominator ab:
[tex]\dfrac{ax\cdot a}{b\cdot a}-\dfrac{bx\cdot b}{a\cdot b}=a^2-b^2 \\\\\\\\ \dfrac{a^2x}{ab}-\dfrac{b^2x}{ab}=a^2-b^2[/tex]
Now that both fractions have the same denominator, we can combine the fractions by subtracting the numerators:
[tex]\dfrac{a^2x-b^2x}{ab}=a^2-b^2[/tex]
Multiply both sides by ab:
[tex]a^2x-b^2x=ab(a^2-b^2)[/tex]
Factor out x:
[tex]x(a^2-b^2)=ab(a^2-b^2)[/tex]
Divide both sides of the equation by (a² - b²) to isolate x:
[tex]x=ab[/tex]
Therefore, the solution to the given equation is:
[tex]\LARGE\boxed{\boxed{x=ab}}[/tex]
