Question

Solve the following equation: \[\dfrac{{\left( {2x - 3} \right)}}{{\left( {2x - 1} \right)}} = \dfrac{{\left( {3x - 1} \right)}}{{\left( {3x + 1} \right)}}\]

Answer 1

Hint: Here we are asked to solve the given equation, that is we have to find the value of the unknown variable \[x\] from the given equation. First, we will try to make the given equation into a simpler form by doing some simplifications. Then if we get a simple equation, we can solve it by the transposition method. That is keeping the unknown variable on one side of the equation and transferring the other terms to the other side of the equation. Then by solving that side we get the required value.

Complete step by step answer:
It is given that \[\dfrac{{\left( {2x - 3} \right)}}{{\left( {2x - 1} \right)}} = \dfrac{{\left( {3x - 1} \right)}}{{\left( {3x + 1} \right)}}\] we aim to solve this equation that is we need to find the value of the variable \[x\].
First, we need to make the given equation into a standard form of an equation by doing some simplifications.
Consider the given equation, \[\dfrac{{\left( {2x - 3} \right)}}{{\left( {2x - 1} \right)}} = \dfrac{{\left( {3x - 1} \right)}}{{\left( {3x + 1} \right)}}\]
On cross multiplying the denominators on both sides, we get
\[\left( {2x - 3} \right)\left( {3x + 1} \right) = \left( {3x - 1} \right)\left( {2x - 1} \right)\]
On simplifying the above equation, we get
\[6{x^2} + 2x - 9x - 3 = 6{x^2} - 3x - 2x + 1\]
On further simplification we get
\[ - 7x - 3 = - 5x + 1\]
Let us simplify it once again.
\[2x + 4 = 0\]
Now let us solve this by the transposition method. First, let us transfer the term \[ + 4\] to the other side by keeping the required term \[2x\] on the left-hand side.
\[2x = - 4\]
On transferring the term \[ + 4\] to the other side its sign will get changed. We aim to find the value of the unknown variable \[x\] so let us take the term \[2\] to the other side. Since it is in multiplication on transferring it will become division on the other side.
\[x = \dfrac{{ - 4}}{2}\]
\[x = - 2\]
Thus, we got the value of the unknown variable as \[x = - 2\].

Note:
 In the above problem, after making the given expression into a simple equation we used the transposition method to find the value of the unknown variable \[x\]. We can also use other methods like the trial-and-error method or the systematic method to solve that equation. In other cases, if we get a quadratic equation, we can solve it by using the standard formula to find the roots of that equation.