There are three cases that can come up as we are solving linear equations. We have already seen one, where an equation has one solution. Sometimes we come across equations that do not have any solutions and even some that have an infinite number of solutions.

Equations with No Solutions

In the example above, a solution was not obtained. Using the the properties of equality to isolate the variable resulted instead in the false statement [latex]4=5[/latex]. Certainly, [latex]4[/latex] is not equal to [latex]5[/latex].

Note that in the second line of the solution above, the statement [latex]2x+4=2x+5[/latex] was obtained after combining like terms on both sides. If we examine that statement carefully, we can see that it was false even before we attempted to solve it. It would not be possible for the quantity [latex]2x[/latex] with [latex]4[/latex] added to it to be equal to the same quantity [latex]2x[/latex] with [latex]5[/latex] added to it. The two sides of the equation do not balance. Since there is no value of x that will ever make this a true statement, we say that the equation has no solution. 

Be careful that you do not confuse the solution [latex]x=0[/latex] with no solution. The solution [latex]x=0[/latex] means that the value [latex]0[/latex] satisfies the equation, so there is a solution. To say that a statement has no solution means that there is no value of the variable, not even [latex]0[/latex], which would satisfy the equation (that is, make the original statement true).

Equations with Many Solutions

You have seen that if an equation has no solution, you end up with a false statement instead of a value for x. It is possible to have an equation where any value for x will provide a solution to the equation. In the example below, notice how combining the terms [latex]5x[/latex] and [latex]-4x[/latex] on the left leaves us with an equation with exactly the same terms on both sides of the equal sign.

When solving, the true statement “[latex]3=3[/latex]” was obtained. When solving an equation reveals a true statement like this, it means that the solution to the equation is all real numbers, that is, there are infinitely many solutions. Try substituting [latex]x=0[/latex] into the original equation—you will get a true statement! Try [latex]x=-\dfrac{3}{4}[/latex]. It will also satisfy the equation. In fact any real value of x will make the original statement true.

Indeed, after combining like terms, the equation [latex]x+3=3+x[/latex] was obtained. It is certainly true that the quantity [latex]x[/latex] with [latex]3[/latex] added to it is equal to [latex]3[/latex] with [latex]x[/latex] added to it by the commutative property of addition.

Watch the following video for demonstrations of equations with no solutions and infinitely many solutions.

The next video demonstrates equations with no or infinitely many solutions involving parentheses.

[Optional] Absolute Value Equations with No Solutions

As we are solving absolute value equations, it is important to be aware of special cases. An absolute value is defined as the distance  of a number from [latex]0[/latex] on a number line, so the absolute value of a number must be a positive. When an absolute value expression is given to be equal to a negative number, we say the equation has no solution (DNE, for short). Notice how this happens in the next two examples.

In this last video, see show more examples of absolute value equations that have no solutions.

We have seen that solutions to equations can fall into three categories:

1. exactly one solution;
2. no solution  (also called DNE for does not exist)); or
3. many solutions (also called infinitely many solutions, or we may say the solution is all real numbers).

Keep in mind that sometimes we do not need to do much algebra to see what the outcome will be.