Question

How do you solve $\dfrac{{7a}}{{3a + 3}} - \dfrac{5}{{4a - 4}} = \dfrac{{3a}}{{2a + 2}}$ ?

Answer 1

Hint: In the given question, we have been asked to find the value of ‘a’ and it is given that $\dfrac{{7a}}{{3a + 3}} - \dfrac{5}{{4a - 4}} = \dfrac{{3a}}{{2a + 2}}$ . To solve this question, we need to get ‘a’ on one side of the “equals” sign, and all the other numbers on the other side. To solve this equation for a given variable ‘a’, we have to undo the mathematical operations such as addition, subtraction, multiplication, and division that have been done to the variables.

Complete step by step solution:
We have
$\dfrac{{7a}}{{3a + 3}} - \dfrac{5}{{4a - 4}} = \dfrac{{3a}}{{2a + 2}}$
We can write this as,
$ \Rightarrow \dfrac{{7a}}{{3(a + 1)}} - \dfrac{5}{{4(a - 1)}} = \dfrac{{3a}}{{2(a + 1)}}$
Taking L.C.M , we will get ,
$ \Rightarrow \dfrac{{28{a^2} - 28a - 15a - 15}}{{12(a + 1)(a - 1)}} = \dfrac{{3a}}{{2(a + 1)}}$
Multiply by $(a + 1)$ to both the side of the equation.
$ \Rightarrow \dfrac{{28{a^2} - 28a - 15a - 15}}{{12(a - 1)}} = \dfrac{{3a}}{2}$
After simplifying like terms, we will get,
$ \Rightarrow \dfrac{{28{a^2} - 43a - 15}}{{12(a - 1)}} = \dfrac{{3a}}{2}$
Multiply by $12(a - 1)$ to both side of the equation , we will get,
$ \Rightarrow \dfrac{{28{a^2} - 43a - 15}}{1} = \dfrac{{3a}}{2} \times 12(a - 1)$
Now, simplify the above equation, we will get,
$ \Rightarrow 28{a^2} - 43a - 15 = 18{a^2} - 18a$
Subtract $18{a^2}$ from both the side of the equation,
$ \Rightarrow 10{a^2} - 43a - 15 = - 18a$
Add $18a$ to both the side of the equation,
$ \Rightarrow 10{a^2} - 25a - 15 = 0$
We can write this as,
$ \Rightarrow 2{a^2} - 5a - 3 = 0$ ,
Or
$
   \Rightarrow 2{a^2} - 6a + a - 3 = 0 \\
   \Rightarrow 2a(a - 3) + 1(a - 3) = 0 \\
   \Rightarrow (2a + 1)(a - 3) = 0 \\
 $
We will get ,
$a = 3$ and $a = - 0.5$ .

Thus the values of ‘a’ are $a = 3$ and $a = - 0.5$ .

Additional information: In the given question, no mathematical formula is being used; only the mathematical operations such as addition, subtraction, multiplication and division is used. Use addition or subtraction properties of equality to gather variable terms on one side of the equation and constant on the other side of the equation. Use the multiplication or division properties of equality to form the coefficient of the variable term equivalent to one.

Note: The important thing to recollect about any equation is that the ‘equals’ sign represents a balance. What the sign says is that what’s on the left-hand side is strictly an equal to what’s on the right-hand side. It is the type of question where only mathematical operations such as addition, subtraction, multiplication and division is used.