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.
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.
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.
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.Let’s look at what one of these equations gives us in the end. 3 + 3x + 5 = x + 2x + 9.
We first combine like terms. On the left side of the equation, we see a 3 and a 5 that can be combined to get 8, and on the right side we see an x and a 2x that can be combined to get 3x. 3x + 8 = 3x + 9.Now I can subtract the 3x from both sides. I get 8 = 9.
Isn’t this interesting? It makes no sense because we know that 8 will never equal 9. They are different numbers and will never equal each other. This means that there is no solution.
Let’s look at a couple more examples. 9 + x = 5 + 4 + x. We first combine like terms: 9 + x = 9 + x.
Next, we can subtract 9 from both sides: x = x. Here we have x = x. Is there just one value that makes this statement true? No, we can actually have any value for x, so this one has infinite solutions.What about the equation 4x + 9 + 1 = 4 + 4 + 2x + 2x? We combine like terms first: 4x + 10 = 8 + 4x.
Now we subtract 4x from both sides: 10 = 8. This doesn’t make sense, so we have no solutions.
Now, let’s review what we’ve learned.
There are two cases in algebra where our answers won’t make sense. The first case is the case of infinite solutions, when all numbers are solutions. The next case is no solutions, when we have no answer.
We can identify which case it is by looking at our results. If we end up with the same term on both sides of the equal sign, such as 4 = 4 or 4x = 4x, then we have infinite solutions. If we end up with different numbers on either side of the equal sign, as in 4 = 5, then we have no solutions.
Review this lesson to learn how to determine if a mathematical equation has an infinite number of solutions or no solution.