Step 1

The given equation is \(\displaystyle{\frac{{{x}-{a}}}{{{x}-{b}}}}={\frac{{{a}}}{{{b}}}}\)

Step 2

Substitute x=0 in given equation,

\(\displaystyle{L}.{H}.{S}={\frac{{{x}-{a}}}{{{x}-{b}}}}\)

\(\displaystyle={\frac{{{0}-{a}}}{{{0}-{b}}}}\)

\(\displaystyle={\frac{{-{a}}}{{-{b}}}}\)

\(\displaystyle={\frac{{{a}}}{{{b}}}}\)

=R.H.S

So, the given value x=0 is a solution of the equation \(\displaystyle{\frac{{{x}-{a}}}{{{x}-{b}}}}={\frac{{{a}}}{{{b}}}}\).

Step 3

Substitute x=b in given equation,

\(\displaystyle{L}.{H}.{S}={\frac{{{x}-{a}}}{{{b}-{b}}}}\)

\(\displaystyle={\frac{{{x}-{a}}}{{{0}}}}\)

we cannot divide by zero, so it is undefined.

So, the given value x=b is not a solution of the equation \(\displaystyle{\frac{{{x}-{a}}}{{{x}-{b}}}}={\frac{{{a}}}{{{b}}}}\).

The given equation is \(\displaystyle{\frac{{{x}-{a}}}{{{x}-{b}}}}={\frac{{{a}}}{{{b}}}}\)

Step 2

Substitute x=0 in given equation,

\(\displaystyle{L}.{H}.{S}={\frac{{{x}-{a}}}{{{x}-{b}}}}\)

\(\displaystyle={\frac{{{0}-{a}}}{{{0}-{b}}}}\)

\(\displaystyle={\frac{{-{a}}}{{-{b}}}}\)

\(\displaystyle={\frac{{{a}}}{{{b}}}}\)

=R.H.S

So, the given value x=0 is a solution of the equation \(\displaystyle{\frac{{{x}-{a}}}{{{x}-{b}}}}={\frac{{{a}}}{{{b}}}}\).

Step 3

Substitute x=b in given equation,

\(\displaystyle{L}.{H}.{S}={\frac{{{x}-{a}}}{{{b}-{b}}}}\)

\(\displaystyle={\frac{{{x}-{a}}}{{{0}}}}\)

we cannot divide by zero, so it is undefined.

So, the given value x=b is not a solution of the equation \(\displaystyle{\frac{{{x}-{a}}}{{{x}-{b}}}}={\frac{{{a}}}{{{b}}}}\).