Question

Explain the 2 different answers you can get when solving an equation and the variable is dropped out

Answer 1

To give a simple answer

if you consider the equation to be that of a line.
then if the variables cancel out, then the two lines will have the same slope.

two lines with the same slope fall under two categories. Parallel lines, and lines on top of each other

parallel lines have no solutions since the lines never cross each other
lines on top of each other have infinitely many solutions  since they "cross" everywhere on the line

Answer 2

Answer with explanation:

⇒Consider an Equation in one variable

  2x+5=4x+3

→When variable is not dropped from any side that is LHS and RHS, we get different Solution

 Taking the variable on one side and constant on another side

→4x-2x=5-3

2 x= 2

Dividing by 2, on both sides

→x=1

→→Now, if we drop variable from both side , we will be equipped with only constants , which may or may not be equal.

→Now if we drop inequality from one side,Suppose LHS

 4x+3=5

4x=5-3

4x=2

$x=\frac{2}{4}\\\\=\frac{1}{2}$

→→If, we drop inequality from other side,that is RHS

5+2x=3

2x=3-5

2x=-2

Dividing by 2, on both sides

x=-1

≡So, we get three distinct solutions ,(a) when variable is not dropped from any side, (b)A variable is dropped from either of LHS and RHS.