Question

How do you solve $2\left( a+3 \right)=12+4a$ ?

Answer 1

Hint: Linear equations of this type can be easily solved by applying the distributive law first. Then we take all the terms associated to $a$ to the left-hand side and the constant terms that do not contain variable $a$ to the right hand-side of the given equation. Further simplifying both the sides of the equation we divide both the sides of the equation by the coefficient of $a$ to get the solution of the given problem.

Complete step-by-step solution:
The linear equation we have is
$2\left( a+3 \right)=12+4a$
We apply the distributive law to the left-hand side of the above equation as shown below
$\Rightarrow 2a+2\times 3=12+4a$
Further simplifying the above equation, we get
$\Rightarrow 2a+6=12+4a$
The above equation can also be written as
$\Rightarrow 12+4a=2a+6$
Now, we must take all the terms related to $a$ to the left-hand side of the equation. To do so we subtract the term $2a$ from both the sides of the above equation as shown below
$\Rightarrow 12+4a-2a=2a+6-2a$
Further simplifying the above equation, we get
$\Rightarrow 12+2a=6$
Now, we must take all the constant terms i.e., terms not associated with $a$ to a to the right-hand side of the equation. To do so we subtract the number $12$ from both the sides of the above equation as shown below
$\Rightarrow 12+2a-12=6-12$
Further simplifying the above equation, we get
$\Rightarrow 2a=-6$
Now, we divide both the sides of the above equation by $2$ as shown below
$\Rightarrow a=\dfrac{-6}{2}$
The above equation can be simplified and written as
$\Rightarrow a=-3$
Therefore, the solution of the given equation $2\left( a+3 \right)=12+4a$ is $a=-3$ .

Note: We must keep in mind that the number that we are multiplying or dividing must be done in both the right- and left-hand side of the equation. Otherwise, the equation will become invalid due to inequality. Also, in this type of calculations and simplifications we must be very careful while subtracting terms from both the sides of the equation, if not done correctly it will get too complicated to solve.