无解方程式

Throughout our math lives, the goal has been to find answers: the value of $x$, the area of a circle, the length of a line. It seems logical, right? Every problem should have a solution that we can write down and circle and feel good about.

Let's remind ourselves of what one of these math problems looks like (and get in a little practice at solving equations while we're at it):

$$\begin{align} 2x + 1 &= 9\\ 2x+1\,\color{#8925AE}{-\,1}&=9\,\color{#8925AE}{-\,1}\\ 2x &= 8\\ \frac{2x}{\color{#8925AE}{2}}&=\frac{8}{\color{#8925AE}{2}}\\ x &= 4 \end{align}$$

And we're done!

The truth is, though, that not all problems work like this. Sometimes, there are equations with no solutions. This means that there is no possible value we could use for $x$ to make the equation true.

Let's take a look at the next equation to start exploring this "no solution" idea. $$5x-3x+6=2x+7-2$$

$$\begin{align} 5x-3x+6 &= 2x+7-2\\ 2x+6 &= 2x+5 \end{align}$$

Take a second to see if any part of that equation looks a little fishy to you. (It's okay if it doesn't—we'll keep explaining below!)

Solve the equation using the same approach as always—get $x$ alone on one side of the equation.

When we actually do that, here's what we're left with: $$\begin{align} 2x+6 &= 2x+5\\ 2x+6\,\color{#8925AE}{-\,6} &= 2x+5\,\color{#8925AE}{-\,6}\\ 2x &= 2x - 1\\ 2x\,\color{#8925AE}{-\,2x} &= 2x\,{\color{#8925AE}{-\,2x}} - 1\\ 0 &= -1 \end{align}$$

This statement (that $0$ equals $-1$) is FALSE.

This is what we mean when we say that an equation "has no solution." The answer we get is something that is not and cannot be true. What happens when an equation is put on a balancing scale, but it has no solutions?

There are two things to be careful about with equations with no solutions:

1. Make sure you didn't make a mistake when solving for $x$. Always check your work!
2. There is a difference between no solution and an equation where the solution is zero. (For instance, for the equation $3x+7=7, x=0$).

Let's practice a little!

$$\begin{align} -1 + 4x + 3 &=-5x - 7 + 9x\\ 4x + 2 &=4x - 7\\ 0 &= -9 \\ \end{align}$$

Once we simplify this equation completely and try to solve for $x$, we get $0=-9$.

That's one of those statements that just can't be true. This means that there is no solution.

Let's try another problem:

Try solving each of those equations. You'll find:

• First choice: $x= \frac{3}{2}$
• Second choice: $0=-4$
• Third choice: $x=-\frac{3}{2}$

The only statement that is false is the second choice. Whenever we end up with a false statement (something that just can't be true), we can say that there is no solution.

Now, let's take a look at the equation $$2x+x+2=3x+2$$

To figure out how many solutions the equation has, we just have to simplify and solve for $x$. Let's do that! $$\begin{align} 2x+x+2&=3x+2\\ 3x+2&=3x+2\\ \cancel{3x-3x}+2&=\cancel{3x-3x}+2\\ 2&=2\\ 0&=0 \end{align}$$ We get the answer $0=0$. This is a true statement always. This means there are infinitely many solutions. Let's look at how we would model that.

We can see there are the same exact things on each side of the scale so it will always be balanced!

We can take this a step further by look at systems of equations, which can also have no solution or infinitely many solutions.

Infinitely many solutions: When both sides are the same

Sometimes, equations might not have a single number as their solution. For example, some have no solutions, and others may have infinitely many.

Having infinitely many solutions means that you couldn't possibly list all the solutions for an equation, because there are infinite. Sometimes that means that every single number is a solution, and sometimes it just means all the numbers that fit a certain pattern.

So how do I know if my equation has infinite solutions?

The same way you would know what the answer was if it had one solution—solve it! Let's try it out on the following equation: $$7(8x + 5) - 35 = 4(14x)$$

$$\begin{align} 7(8x + 5) - 35 &= 4(14x)\\ 56x + 35 - 35 &= 56x\\ 56x &= 56x \end{align}$$

Both sides of the equation simplify out to the same term! Since both sides are now identical, you won't be able to do anything to make them not identical since all operations have to be done to both sides of an equation. This means that any solution will work to make these two equations identical, so the equation has infinitely many solutions.

(Videos) Linear Equations with No Solutions or Infinitely Many

by MySecretMathTutor

This video by MySecretMathTutor works through a problem that has no solution.

$\large{\text{Summary}}$

Normally when solving problems you end up with something at the end saying, $x=$ [some number]. When a problem has no solution you'll end up with a statement that's false. For example: $$0=1$$ This is false because we know zero can't equal one. Therefore we can conclude that the problem has no solution.

The problem that he looks at in the video is: $$13x-5(x-4)=8x+5$$

You can solve this as you would any other equation. $$\begin{align} 13x-5(x-4)&=8x+5\\ 13x-\color{#8925AE}{5x+20}&=8x+5\\ 8x+20&=8x+5\\ 8x\,{\color{#8925AE}{-\,8x}}+20&=8x\,{\color{#8925AE}{-\,8x}}+5\\ 20&=5 \end{align}$$

We can see that we used the distributive property, we combined like terms, and we solved by grouping to get our solution.

We see this is a false statement because $20$ does not equal $5$. So, we say this problem has no solution. This means no matter what number you put in for the variable $x$, you will never get anything that equals one another.

(Video) Infinitely many solutions

by MySecretMathTutor

This video by MySecretMathTutor shows you an example of a problem that has an infinite number of solutions.

$\large{\text{Summary}}$

When a problem has infinite solutions, you'll end up with a statement that's true no matter what. For example: $$3=3$$ This is true because we know $3$ equals $3$, and there's no variable in sight. Therefore we can conclude that the problem has infinite solutions.

The problem that he looks at in the video is: $$4(2x-8)=8(x-4)$$

You can solve this as you would any other equation. $$\begin{align} 4(2x-8)&=8(x-4)\\ \color{#8925AE}{8x-32}&=\color{#8925AE}{8x-32}\\ 8x\,{\color{#8925AE}{-\,8x}}-32&=8x\,{\color{#8925AE}{-\,8x}}-32\\ -32&=-32 \end{align}$$ So we see this is a true statement because $-32$ does equal $-32$, and will no matter what you plug in for $x$. So we say this problem has infinite solutions. This means no matter what number you put in for the variable $x$, you will always get a number equaling itself.

$$\begin{align} (3x + 12) + 7 &= 3(x + 4) - 13\\ 3x + 19 &= 3x + 12 - 13\\ 3x + 19 &= 3x - 1\\ 19 &= 1\\ \end{align}$$

If we simplify the expression as much as possible, we get $19 = -1$. We know this isn't true, so the equation has no solutions.