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Question

Question: How do you solve \(\dfrac{3}{4x}=\dfrac{5}{x+2}\)?...

How do you solve 34x=5x+2\dfrac{3}{4x}=\dfrac{5}{x+2}?

Explanation

Solution

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 xx. 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 34x=5x+2\dfrac{3}{4x}=\dfrac{5}{x+2} is a linear equation of xx.
We apply cross-multiplication to multiply (x+2)\left( x+2 \right) with 3 and 4x4x with 5.

& \dfrac{3}{4x}=\dfrac{5}{x+2} \\\ & \Rightarrow 3\left( x+2 \right)=4x\times 5 \\\ \end{aligned}$$ We complete the multiplication to get $$3x+6=20x$$. We take all the variables and the constants on one side and get $$3x+6-20x=0$$. There are three variables which are $3x,20x$. The binary operation between them is subtraction which gives us $3x-20x=-17x$. Now we take the constants. There is one such constant which is 6. The final solution becomes $\begin{aligned} & 6-17x=0 \\\ & \Rightarrow -17x=-6 \\\ \end{aligned}$. Now we divide with $-17$ to get $$\begin{aligned} & -17x=-6 \\\ & \Rightarrow x=\dfrac{-6}{-17}=\dfrac{6}{17} \\\ \end{aligned}$$ 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.