*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!

You should understand how to work with multi-step equations and solve for them before beginning this lesson.

If you have not yet completed that lesson, found in the right-hand sidebar under **Related Lessons**, do so first.

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, you will instead try 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. 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 the equation would look like if you 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!

You should have:

2x |
+ | 2 | = | 4 | + | x |

First, 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 you solve an equation, you 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*!

Take the x from the right side and subtract on both sides of the equation to get:

x |
+ | 2 | = | 4 |

From here, you just have to subtract 2 from both sides to get:

x |
= | 2 |

To check your work, just take the x value and plug it back into the original formula:

2(2) | + | 2 | = | 4 | + | 2 | |||

6 | = | 6 |

This is true, so you are right!

Now that you know how to work through an equation, kick it up a notch! 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 this equation as an example:

4 | - | 5x |
= | 4 | - | 5x |

When you try to bring all the variables to the left side, you get:

4 | - | 5x |
+ | 5x |
= | 4 |

Simplify to get:

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. Another way to name i is that all real numbers are solutions.

Let's look at another equation:

7x |
+ | 4 | = | 7x |
+ | 8 |

When solved, it is:

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 in the *Got It?* section.