Question

Simplify $\dfrac{{5{\text{y}} + 3}}{{{\text{y}} + 2}} - \dfrac{{2{\text{y}} - 3}}{{{\text{y}} + 2}}$ ?

Answer 1

Hint: In this question, they have given two fractions involving variables and asked to find the difference between them. First we have to check both the denominators, since the denominators are the same for both the terms; we can simply subtract the second term in the numerator from the first term in the numerator and simplify it to find the correct answer.

Complete step-by-step solution:
The given question is $\dfrac{{5{\text{y}} + 3}}{{{\text{y}} + 2}} - \dfrac{{2{\text{y}} - 3}}{{{\text{y}} + 2}}$,
Since the denominators of both the terms are the same, we can directly subtract the numerator.
So, $\dfrac{{5{\text{y}} + 3}}{{{\text{y}} + 2}} - \dfrac{{2{\text{y}} - 3}}{{{\text{y}} + 2}}$
On rewriting we get,
$ \Rightarrow \dfrac{{\left( {5{\text{y}} + 3} \right) - \left( {2{\text{y}} - 3} \right)}}{{{\text{y}} + 2}}$
Removing the bracket, the minus sign will be multiplied to the second term and that will make the signs turn to opposite signs,
$ \Rightarrow \dfrac{{5{\text{y}} + 3 - 2{\text{y}} + 3}}{{{\text{y}} + 2}}$
By separating the terms with variable and numbers we can easily solve,
$ \Rightarrow \dfrac{{5{\text{y}} - 2{\text{y}} + 3 + 3}}{{{\text{y}} + 2}}$
On simplify the term and we get,
$ \Rightarrow \dfrac{{3{\text{y}} + 6}}{{{\text{y}} + 2}}$
Now we can see that $3$ is common in numerator.
So we have to take $3$ out, it we take $3$ from $3{\text{y}} + 6$, we will get $\left( {{\text{y}} + 2} \right)$ in the numerator.
Now we can cancel the $\left( {{\text{y}} + 2} \right)$ on both numerator and denominator to simplify it and get the answer,
$ \Rightarrow \dfrac{{3\left( {{\text{y}} + 2} \right)}}{{{\text{y}} + 2}}$
On cancel the term and we get
$ \Rightarrow 3$

Therefore, $3$ is the required answer.

Note: In this sum we directly subtracted the numerator because the denominators are the same. If the denominators were different in the given terms, we need to find a new denominator by cross multiplying the terms. For example
If the question is $\dfrac{{2{\text{x}} + 1}}{{{\text{x}} - 1}} + \dfrac{{3{\text{x}}}}{{{\text{x}} + 2}}$ then we have to find new denominator since they are different.
$ = \dfrac{{\left( {2{\text{x}} + 1} \right)\left( {{\text{x}} + 2} \right) + 3{\text{x}}\left( {{\text{x}} - 1} \right)}}{{\left( {{\text{x}} - 1} \right)\left( {{\text{x}} + 2} \right)}}$
So here $\left( {{\text{x}} - 1} \right)\left( {{\text{x}} + 2} \right)$ i.e., \[{{\text{x}}^2} + 2{\text{x}} - {\text{x}} - 2 \Rightarrow {{\text{x}}^{\text{2}}} + {\text{x}} - 2\]will be the denominator and we have to proceed with the numerator.
Often students will make mistakes by directly adding or subtracting the given terms in the fraction which is completely wrong.