Question

How do you solve $\dfrac{3}{4x}=\dfrac{5}{x+2}$?

Answer 1

Hint: We solve the given linear equation by simplifying the equation. We cross-multiply the equations. Then we apply the binary operation of division to get the value of $x$. We use the G.C.D of the denominator and the numerator to divide both of them. We get the simplified form when the G.C.D is 1.

Complete step by step solution:
The given equation $\dfrac{3}{4x}=\dfrac{5}{x+2}$ is a linear equation of $x$.
We apply cross-multiplication to multiply $\left( x+2 \right)$ with 3 and $4x$ with 5.
\[\begin{align}
  & \dfrac{3}{4x}=\dfrac{5}{x+2} \\
 & \Rightarrow 3\left( x+2 \right)=4x\times 5 \\
\end{align}\]
We complete the multiplication to get \[3x+6=20x\].
We take all the variables and the constants on one side and get \[3x+6-20x=0\].
There are three variables which are $3x,20x$.
The binary operation between them is subtraction which gives us $3x-20x=-17x$.
Now we take the constants. There is one such constant which is 6.
The final solution becomes
$\begin{align}
  & 6-17x=0 \\
 & \Rightarrow -17x=-6 \\
\end{align}$.
Now we divide with $-17$ to get
\[\begin{align}
  & -17x=-6 \\
 & \Rightarrow x=\dfrac{-6}{-17}=\dfrac{6}{17} \\
\end{align}\]
Therefore, the solution is $x=\dfrac{6}{17}$.

Note: Simplified form is achieved when the G.C.D of the denominator and the numerator is 1. This means we can’t eliminate any more common root from them other than 1. Therefore, $x=\dfrac{6}{17}$ is in its simplest form.