Question

How do you solve $2(3x + 4) = 5(x - 5) + 1 ?$

Answer 1

Hint: : In this problem, firstly simplify the left hand side by multiplying 2 to $3x + 4$. Also simplify the right hand side by multiplying 5 to $x - 5$. Then try to bring the terms containing the variable x to the left hand side and take the constant terms to the right hand side. Then solve it to obtain the value of x. Also substitute back the value of x in the given equation and obtain the left hand side is equal to right hand side.

Complete step by step solution:
Given $2(3x + 4) = 5(x - 5) + 1$ ……(1)
We need to simplify the above equation to obtain the value of x.
Firstly we simplify the left hand side.
Multiply 2 to $3x + 4$, we obtain the equation in simplified form.
$2(3x + 4) = 2(3x) + 2(4)$
$ \Rightarrow 2(3x + 4) = 6x + 8$
Now we simplify the right hand side in a similar manner.
Multiply 5 to $x - 5$, we obtain the equation in simplified form.
$5(x - 5) = 5(x) + 5( - 5)$
$ \Rightarrow 5(x - 5) = 5x - 25$
Now we substitute the simplified forms obtained above in the equation (1) and then we further simplify it to obtain the value of x.
Substituting in equation (1), we get,
 $2(3x + 4) = 5(x - 5) + 1$
$ \Rightarrow 6x + 8 = 5x - 25 + 1$
Add +1 to -25 which gives us -24.
So we must have the equation as,
$6x + 8 = 5x - 24$
Now subtract $5x$ from both the sides to get the variable x on the same side i.e. left hand side of the equation.
So we must have the equation as,
$6x + 8 - 5x = 5x - 5x - 24$
$ \Rightarrow 6x - 5x + 8 = 0 - 24$
$ \Rightarrow x + 8 = - 24$
Now we subtract 8 from both sides of the equation to keep constant terms on the same side i.e. right hand side of the equation.
So we must have the equation as,
$x + 8 - 8 = - 24 - 8$
$ \Rightarrow x + 0 = - 32$
$ \Rightarrow x = - 32$
Now we substitute the value of x i.e. $x = - 32$ in the equation (1) to get the left hand side equal to the right hand side.
Substituting $x = - 32$ in the equation (1), we get,
$2(3x + 4) = 5(x - 5) + 1$
$2(3(- 32) + 4) = 5((- 32) - 5) + 1$
$ \Rightarrow 2(- 96 + 4) = 5(- 37) + 1$
$ \Rightarrow 2(- 96) + 2(4) = - 185 + 1$
$ \Rightarrow - 192 + 8 = - 184$
$ \Rightarrow - 184 = - 184$
i.e. Left hand side = Right hand side.

Therefore $x = - 32$ is the required value of the variable x.

Note: Here we must note that the variable x takes on different numerical values but the constant remains unchanged. An equation remains unchanged if its left hand side and right hand side are interchanged.
It is important to know the following basic facts.
An equation remains unchanged or undisturbed if it satisfies the following conditions.
(1) If L.H.S. and R.H.S. are interchanged.
(2) If the same number is added on both sides of the equation.
(3) If the same number is subtracted on both sides of the equation.
(4) When both L.H.S. and R.H.S. are multiplied by the same number.
(5) When both L.H.S. and R.H.S. are divided by the same number.