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Solving Equations with Infinite Solutions or No Solutions

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In algebra, there are two scenarios that give us interesting results. Watch this video lesson to learn how you can distinguish problems that have no answers and problems that have an infinite number of answers.
Solving Equations
When you’re solving equations in algebra, it is kind of like a treasure hunt. You are looking for your x. You want to know where your x is, so you can go find your treasure. With most equations, you will get an answer letting you know where your treasure is located. For example, solving the equation x + 3 = 4 by subtracting 3 from both sides gives us x = 1 as our answer and location of our treasure.

But sometimes, an equation that you are trying to solve for gives you an answer that just doesn’t make sense. It is these types of answers that we are going to discuss in this video lesson. It is important to understand these so you can spot them and identify the equations as unsolvable because they have an answer that doesn’t make sense. We will go over the two possible cases where the answer doesn’t make sense.

Infinite Solutions
The first is when we have what is called infinite solutions. This happens when all numbers are solutions. This situation means that there is no one solution. In terms of our treasure hunt, it means that we can’t find the treasure because the location of the treasure can be anywhere. There is no x that marks the spot. Our x here marks the whole world, which doesn’t help us.

The equation 2x + 3 = x + x + 3 is an example of an equation that has an infinite number of solutions. Let’s see what happens when we solve it. We first combine our like terms. We see two x terms that we can combine to make 2x.

2x + 3 = 2x + 3.

Now we can subtract 3 from both sides: 2x = 2x. Hmm. This is an interesting situation; both sides are equal to each other. How many different values of x will make this equation true? Why, isn’t it any number? Yes, and so we have our infinite solutions.

No Solutions
The next case is what is called no solutions. In this case, we have no answer. Our problem equation is a dud. In terms of helping us find our treasure, it actually leads us down the wrong path, to a dead end, so to speak. We think we are going somewhere, but in the end, this equation just laughs at us with an end that doesn’t make sense.

 
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