Question

How do you solve \[7 - 3x\, = \,2x + 7 - 5x\]?

Answer 1

Hint: Rearrange all the terms and make the equation equal to zero. By putting all the terms on one side and making it equal to 0 will make it easy to solve. We can put together the like terms on one side and shift the non like terms to the other side also for easy and quick solving.

Complete step by step solution:
According to the question, the equation is:
\[7 - 3x\, = \,2x + 7 - 5x\]
First, we will try to arrange the terms in such a manner where \[x\] comes in priority so that later it will be easy to solve.
\[ \Rightarrow - 3x + 7\, = \,2x + 7 - 5x\]
Now, we need to make groups of like terms and unlike terms for easy solving.
\[ \Rightarrow - 3x + 7\, = \,2x - 5x + 7\]
As you can see, when we are solving the equation, both the terms on opposite sides are equal. So, when we shift all the terms to one side and make it equal to 0, then all the terms will cancel each other.
\[ \Rightarrow - 3x + 3x + 7 - 7\, = \,0\]
\[ \therefore 0\, = \,0\]
There is no value for the variable \[x\]. From this answer we can say that the equation has an infinite number of solutions.

Additional information:
By solving, we found the result, \[- 3x + 7\, = \, - 3x + 7\], and this shows that both the sides are equal. When both sides are equal in a problem, then the problem is having an infinite number of solutions and the problem is an identity and all the real numbers are its solutions.

Note: The above method is very easy to solve, but there is also a method to solve it. We can add or subtract the same terms respectively which are on either side so that the terms get canceled out, and we are finally left with the main terms which are mainly the \[x\] terms which we want to solve. Like in the equation \[ - 3x + 7\, = \, - 3x + 7\], we can subtract \[7\] from both sides so that it gets canceled.