Question

Solve for $x$ $\dfrac{2-7x}{1-5x}=\dfrac{3+7x}{4+5x}$
A) $\dfrac{3}{2}$
B) $\dfrac{1}{2}$
C) $\dfrac{3}{4}$
D) $\dfrac{1}{4}$

Answer 1

Hint: To solve this type of linear equation we will take variable terms to one side and constants to the other side then we will apply different mathematical operations to get the required solution.

Complete answer:
Our given equation is
$\dfrac{2-7x}{1-5x}=\dfrac{3+7x}{4+5x}$
First we will use cross multiplication to simplify this
$\begin{align}
  & \dfrac{2-7x}{1-5x}=\dfrac{3+7x}{4+5x} \\
 & \Rightarrow \left( 2-7x \right)\left( 4+5x \right)=\left( 3+7x \right)\left( 1-5x \right) \\
\end{align}$
Now we will simplify both left hand side and right hand side
First we will simplify left hand side
$\begin{align}
  & \left( 2-7x \right)\left( 4+5x \right) \\
 & \Rightarrow 2\times 4-4\times 7x+2\times 5x-7x\times 5x \\
 & \Rightarrow 8-28x+10x-35{{x}^{2}} \\
 & \Rightarrow 8-18x-35{{x}^{2}} \\
\end{align}$
We can also write it as
$-35{{x}^{2}}-18x+8$
Now we will solve right hand side
$\begin{align}
  & \left( 3+7x \right)\left( 1-5x \right) \\
 & \Rightarrow 3\times 1-3\times 5x+7x\times 1-7x\times 5x \\
 & \Rightarrow 3-15x+7x-35{{x}^{2}} \\
 & \Rightarrow 3-8x-35{{x}^{2}} \\
\end{align}$
We can also write it as
$-35{{x}^{2}}-8x+3$
So now our equation becomes
$-35{{x}^{2}}-18x+8=-35{{x}^{2}}-8x+3$
This is a quadratic equation but if we simplify this we will get a linear equation.
Now we will try to shift all variable terms on left hand side and all constants to right hand side
$\begin{align}
  & -35{{x}^{2}}-18x+8=-35{{x}^{2}}-8x+3 \\
 & \Rightarrow -35{{x}^{2}}-18x+35{{x}^{2}}+8x=3-8 \\
\end{align}$
Now we will simplify this
$\begin{align}
  & -35{{x}^{2}}-18x+35{{x}^{2}}+8x=3-8 \\
 & \Rightarrow -18x+8x=-5 \\
 & \Rightarrow -10x=-5 \\
\end{align}$
$\begin{align}
  & \Rightarrow x=\dfrac{-5}{-10} \\
 & \Rightarrow x=\dfrac{-1}{-2} \\
 & \Rightarrow x=\dfrac{1}{2} \\
\end{align}$
$\therefore x=\dfrac{1}{2}$ is our required answer.

So, the correct answer is “Option B”.

Note:
There are three methods to solve linear equations in one variable.
1. Trial and Error.
2. Inverse operations.
3. Transposition Method.