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 don’t have any solutions, and even some that have an infinite number of solutions. The case where an equation has no solution is illustrated in the next examples.

Equations with no solutions

This is not a solution! You did not find a value for [latex]x[/latex]. Solving for [latex]x[/latex] the way you know how, you arrive at the false statement [latex]4=5[/latex]. Surely  [latex]4[/latex] cannot be equal to [latex]5[/latex]!

This may make sense when you consider the second line in the solution where like terms were combined. If you multiply a number by [latex]2[/latex] and add [latex]4[/latex], you would never get the same answer as when you multiply that same number by [latex]2[/latex] and add [latex]5[/latex]. Since there is no value of [latex]x[/latex] that will ever make this a true statement, the solution to the equation above is “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. “No solution” means that there is no value, not even [latex]0[/latex], which would satisfy the equation.

Also, be careful not to make the mistake of thinking that the equation [latex]4=5[/latex] means that [latex]4[/latex] and [latex]5[/latex] are values for [latex]x[/latex] that are solutions. If you substitute these values into the original equation, you’ll see that they do not satisfy the equation. This is because there is truly no solution—there are no values for [latex]x[/latex] that will make the equation [latex]12+2x–8=7x+5–5x[/latex] true.

Algebraic Equations with an Infinite Number of Solutions

You have seen that if an equation has no solution, you end up with a false statement instead of a value for [latex]x[/latex]. It is possible to have an equation where any value for [latex]x[/latex] 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.

You arrive at the true statement “[latex]3=3[/latex].” When you end up with a true statement like this, it means that the solution to the equation is “all real numbers.” Try substituting [latex]x=0[/latex] into the original equation—you will get a true statement! Try [latex]x=-\frac{3}{4}[/latex], and it will also check to be a true statement!

This equation happens to have an infinite number of solutions. Any value for [latex]x[/latex] that you can think of will make this equation true. When you think about the context of the problem, this makes sense—the equation [latex]x+3=3+x[/latex] means “some number plus [latex]3[/latex] is equal to [latex]3[/latex] plus that same number.” We know that this is always true—it’s the commutative property of addition!

In the following video, we show more examples of attempting to solve a linear equation with either no solution or many solutions.

In this video, we show more examples of solving linear equations with either no solutions or many solutions.

In the following video, we show more examples of solving linear equations with parentheses that have either no solution or many solutions.

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 from [latex]0[/latex] on a number line, so it must be a positive number. When an absolute value expression is equal to a negative number, we say the equation has no solution, or DNE. Notice how this happens in the next two examples.

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

Summary

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

• One solution.  This is when you find the only value of the variable, such as [latex]x = 5[/latex].
• No solution, DNE (does not exist).  This is when a false statement appears, like [latex]4 = 7[/latex].
• Many solutions, also called infinitely many solutions or All Real Numbers.  This is when a true statement appears, like [latex]x + 3 = x + 3[/latex].

And sometimes, we don’t need to do much algebra to see what the outcome will be.