When solving an equation, the goal is to find the values of the variables that make the equation true. However, in some cases, when you manipulate the equation, the variables can cancel out entirely, leading to a statement that contains only constants, such as \(4 = 4\).
When you end up with a statement like \(4 = 4\), it means that the original equation is true for all values of the variable(s) involved. In other words, the equation is an identity. This indicates that there are infinitely many solutions to the equation, because any input (value for the variable) will satisfy the equation.
For example, consider the equation:
\[ 2x + 2 = 2(x + 1) \]
If you simplify both sides, you eventually find:
\[ 2x + 2 = 2x + 2 \]
When you subtract \(2x\) from both sides, you obtain:
\[ 2 = 2 \]
This is a true statement and indicates that the equation holds for any value of \(x\). Therefore, in this case, \(x\) can be any real number, and the solution set is all real numbers.
In contrast, if you end up with a statement like \(4 = 5\), this would indicate that there are no solutions, as this statement is false.
So, to summarize:
- If you solve to get a true statement like \(4 = 4\), the equation has infinitely many solutions.
- If you get a false statement, then the equation has no solutions at all.