Question

Solve the following equation: $\dfrac{x-1}{7x-14}=\dfrac{x-3}{7x-26}$.
(a) 13
(b) 8
(c) 17
(d) 6

Answer 1

Hint: We know that for any given equation, we can perform any algebraic operation like addition, subtraction, multiplication and division, on both sides of this equation simultaneously. Using this concept, we can simplify this equation to get the value of $x$.

Complete step by step answer:
We are given the following equation in the variable $x$,
$\dfrac{x-1}{7x-14}=\dfrac{x-3}{7x-26}$
We need to solve this equation, and this means that we have to find the value of variable $x$.
Let us first multiply both sides of this equation with the expression $\left( 7x-14 \right)$. Hence, we get
$\dfrac{x-1}{7x-14}\times \left( 7x-14 \right)=\dfrac{x-3}{7x-26}\times \left( 7x-14 \right)$
We can easily simplify the right hand side of this equation and cancel the terms on the left hand side to this equation, to get
$x-1=\dfrac{7{{x}^{2}}-21x-14x+42}{7x-26}$
Or, we can write this as,
$x-1=\dfrac{7{{x}^{2}}-35x+42}{7x-26}...\left( i \right)$
Let us now multiply the expression $\left( 7x-26 \right)$ on both sides of equation (i). Thus, we get
$\left( x-1 \right)\times \left( 7x-26 \right)=\dfrac{7{{x}^{2}}-35x+42}{\left( 7x-26 \right)}\times \left( 7x-26 \right)$
Now, we again the multiply the two terms on the left hand side, and can cancel the terms on the right hand side. Thus, we now have
$7{{x}^{2}}-26x-7x+26=7{{x}^{2}}-35x+42$
We can also write the above equation as
$7{{x}^{2}}-33x+26=7{{x}^{2}}-35x+42...\left( ii \right)$
Let us now subtract both sides of this equation by $7{{x}^{2}}$.
$\left( 7{{x}^{2}}-33x+26 \right)-7{{x}^{2}}=\left( 7{{x}^{2}}-35x+42 \right)-7{{x}^{2}}$
Hence, we can simplify this equation as
$-33x+26=-35x+42$
Add $35x$ on both sides of the above equation, we get
$\left( -33x+26 \right)+35x=\left( -35x+42 \right)+35x$
Thus, we now have
$2x+26=42$
On subtracting both sides of this equation by 26, we can write this equation as
$\left( 2x+26 \right)-26=42-26$
We can simplify the above equation as
$2x=16$
On dividing both sides of the above equation by 2, we get
$\dfrac{2x}{2}=\dfrac{16}{2}$
Hence, we get
$x=8$
Thus, the solution of $\dfrac{x-1}{7x-14}=\dfrac{x-3}{7x-26}$ is $x=8$.

So, the correct answer is “Option b”.

Note: We can directly arrive at equation (ii) by using the concept of cross multiplication, according to which if $\dfrac{A}{B}=\dfrac{C}{D}$, then we can also write this equation as $AD=BC$.Also, we must take care to always perform any operation on both sides of the equation, and not just one.