Question

How do you solve $\dfrac{3x+2}{3x-2}=\dfrac{4x-7}{4x+7}$?

Answer 1

Hint: In this problem we need to solve the given equation. We can observe that the equation contains a fraction in both LHS and RHS so we will first do cross multiplication and use multiplication distribution law to simplify it. After that we will perform some arithmetic operation based on the equation, we have to solve the equation.

Complete step by step answer:
Given that, $\dfrac{3x+2}{3x-2}=\dfrac{4x-7}{4x+7}$.
Cross multiplying the above fractions, then we will get
$\Rightarrow \left( 3x+2 \right)\left( 4x+7 \right)=\left( 4x-7 \right)\left( 3x-2 \right)$
Using distribution law of multiplication in the above equation, then we will have
$\begin{align}
  & \Rightarrow 3x\left( 4x \right)+2\left( 4x \right)+3x\left( 7 \right)+2\left( 7 \right)=4x\left( 3x \right)-7\left( 3x \right)+4x\left( -2 \right)-7\left( -2 \right) \\
 & \Rightarrow 12{{x}^{2}}+8x+21x+14=12{{x}^{2}}-21x-8x+14 \\
\end{align}$
Simplifying the above equation, then we will get
$\Rightarrow 12{{x}^{2}}+29x+14=12{{x}^{2}}-29x+14$
In the above equation we have $12{{x}^{2}}$, $14$ on both sides. So, they will be get cancelled, then the above equation is modified as
$\Rightarrow 29x=-29x$
Dividing the above equation with $29$ on both sides, then we will get
$\Rightarrow \dfrac{29x}{29}=-\dfrac{29x}{29}$
We know that $\Rightarrow \dfrac{a}{a}=1$. Using this rule in the above equation, then we will have
$\Rightarrow x=-x$
Adding $x$ on both sides of the above equation, then we will get
$\Rightarrow x+x=-x+x$
We know that $-a+a=0$, from this rule we can write the above equation as
$\Rightarrow 2x=0$
Again, dividing the above equation with $2$, then we will get
$\begin{align}
  & \Rightarrow \dfrac{2x}{2}=\dfrac{0}{2} \\
 & \Rightarrow x=0 \\
\end{align}$

Hence the solution of the given equation $\dfrac{3x+2}{3x-2}=\dfrac{4x-7}{4x+7}$ is $x=0$.

Note: For this problem we can also follow another method to solve the given equation. We will consider both LHS and RHS individually and we will rationalize the both the fractions. To rationalize the fraction $\dfrac{1}{ax+b}$ we will multiply and divide the fraction with $ax+b$. So, we will follow this procedure and rationalize the both the fraction by using the algebraic formulas. After simplifying the equation, we will apply arithmetic operations to solve the equation.