*Contributor: Mason Smith. Lesson ID: 11187*

There are two sides to every story, but when there are variables on both sides of an equation, that's a different animal! Learn about one-solution, many-solution, and even no-solution equations!

categories

subject

Math

learning style

Visual

personality style

Lion

Grade Level

Middle School (6-8), High School (9-12)

Lesson Type

Quick Query

Q: Why did the variable add its opposite?

A: To get to the other side!

We have the basic idea of how to work with multi-step equations and solve for them from the previous lesson, found in the right-hand sidebar under **Related Lessons**, but the next step is to solve equations that are more difficult and don't always have the variables together.

Instead of trying to figure out just one side of the equation to get the variable by itself, we will instead be trying to get the same variable from both sides of the equal sign onto one side, and then solve for the variable.

Look at the algebra scale below and, knowing that all the bags contain the same amount of candy, try to guess how much candy is in each bag or, in math-speak, what the variable is:

Did you guess that each bag has two candies? How do you think our equation would look like if we swapped out the candies and bags for numbers and variables? Remember the fulcrum — or tipping point — of the scale is where your equal sign would be!

I came up with 2*x* + 2 = 4 + *x*.

First, we have to get the bags — or variables — on one side of the equation, and all the loose candy — or numbers — on the other side of the equation. Remember, whenever we solve an equation, we have to work on both sides at the same time.

**Hint**: When you are solving for x, always strive to get the x on the side where x is *positive* and *bigger*, so *move the x first*!

We will take the x from the right side and subtract on both sides of the equation to get *x *+ 2 = 4. From here, we just have to subtract 2 from both sides to get *x = *2.

If we want to check our work, we just take our x value and plug it back into the original formula so, 2(2) + 2 = 4 + 2 or 6 = 6, which is true. We are right!

Now that we know how to work through an equation, we are going to kick it up a notch and figure out how to solve a more complicated equation that doesn't have just one solution but many, or sometimes no solution at all!

Take for an example the equation 4 – 5*x* = 4 – 5*x**.* When we try to bring all our variables to the left side, we get 4 – 5*x *+ 5*x* = 4, which, when we simplify, gives us 4 = 4. Now this is where many people make mistakes, so pay close attention to the numbers that you have left:

If the equation is *true* and makes sense, then there are infinitely *many *solutions.

If the equation is *false* and makes no sense, then there are *no* solutions.

If the equation is in the form variable (x) = 7 or some other number, there is *one* solution.

Let's go back to our solved equation 4 = 4; are 4 and 4 the same thing? Yes! So there are infinitely *many* solutions, or, another way to name it, all real numbers are solutions.

Let's look at another equation, say 7*x *+ 4 = 7*x *+ 8, which, when solved, gives us 4 = 8. Does this make much sense? Are 4 and 8 the same thing? No! Therefore, there are *no* solutions.

Now that you have an idea of how to solve linear equations with variables on both sides, practice the different methods of solving.

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