Question

How many solutions does $2\left( {3x - 4} \right) = - \left( {8 - 6x} \right)$?

Answer 1

Hint: To solve this question, we need to know the basic theory related to the Identities and equations with its solutions. If solving an equation yields a statement that is true for a single value for the variable then the equation has one solution. But If solving an equation yields a statement that is always true, then in this case we will say the equation has infinite solutions as briefly discussed below.

Complete step-by-step solution:
Given that the equation as given below:
$ \Rightarrow $$2\left( {3x - 4} \right) = - \left( {8 - 6x} \right)$
Expanding the left-hand side of the above equation as given below:
$ \Rightarrow $$2\left( {3x} \right) - 2\left( 4 \right) = - \left( {8 - 6x} \right)$
The left-hand side of the above expression is expanded and multiplied by the number 2 with $\left( {3x - 4} \right)$.
And we will get,
$ \Rightarrow $$6x - 8 = - \left( {8 - 6x} \right)$
Now the above obtained equation is arranged in such a way that all the like terms are the unlike terms are grouped together-
$ \Rightarrow $$\left( {6x - 6x} \right) = - 8 + 8$
Once again, the x terms cancel out, but we are left with a true statement. -8 does equal 8, and will no matter what we put in for x.
$ \Rightarrow $ $0 = 0$
This is a TRUE statement, but there is no x term left to solve for.
Therefore, this represents an identity, which means it is true for any value of the variable.
Therefore, we can conclude that there are infinitely many solutions. Whatever number we put in for the variable x, it'll always give you a true statement.

Note: Always remember that If the equation ends with a false statement (ex: 0 = 3) then you know that there's no solution but If the equation ends with a true statement (ex: 2 = 2) then you know that there's infinitely many solutions or all real numbers.