Question

How do you solve $\dfrac{3}{{2x - 1}} - \dfrac{4}{{3x - 1}} = 1$?

Answer 1

Hint: We will first take the least common multiple of the left hand side and then just find the resulting fraction and then cross – multiply it to find a new equation and thus solve it.

Complete step-by-step solution:
We are given that we are required to solve $\dfrac{3}{{2x - 1}} - \dfrac{4}{{3x - 1}} = 1$.
Let us take the least common multiple of the above equation to obtain the following equation:-
$ \Rightarrow \dfrac{{3(3x - 1) - 4(2x - 1)}}{{(2x - 1)(3x - 1)}} = 1$
Simplifying the numerator on the left hand side of the above equation to obtain the following equation:-
$ \Rightarrow \dfrac{{9x - 3 - 8x + 4}}{{(2x - 1)(3x - 1)}} = 1$
Simplifying the numerator further on the left hand side of the above equation to obtain the following equation:-
$ \Rightarrow \dfrac{{x + 1}}{{(2x - 1)(3x - 1)}} = 1$
Simplifying the denominator on the left hand side as well in the above equation to obtain the following equation:-
$ \Rightarrow \dfrac{{x + 1}}{{6{x^2} - x + 1}} = 1$
Cross – multiplying the terms in the above equation to obtain the following equation:-
$ \Rightarrow x + 1 = 6{x^2} - x + 1$
Re – arranging the terms on one side in the above expression to get the following expression:-
\[ \Rightarrow 6{x^2} - x + 1 - x - 1 = 0\]
Simplifying the above equation by clubbing the right terms of the above equation ( take x and 1 from addition in the left hand side to subtraction in the right hand side)to obtain the following equation:-
\[ \Rightarrow 6{x^2} - 2x = 0\]
We can take out 2x common from it to get the following expression:-
\[ \Rightarrow 2x(3x - 1) = 0\]
Now, we have roots as either $2x = 0$ which implies that $x = 0$ or we have $3x – 1 = 0$ which implies that $x = \dfrac{1}{3}$.

Hence, the answer is 0 or $\dfrac{1}{3}$.

Note: The students must note that we have used a theorem in the last step which states that:
If a.b = 0, then either a = 0 or b = 0.
Thus, we obtained the roots of the given equation.
Here, if we replace a by 2x and b by (3x – 1), we get the same situation as we faced earlier in solution above.