Solving algebra equations is difficult for many. If you struggle with algebra solving equations can be improved using this lesson and practice.
In the previous section we talked about simplifying expressions. In this section we'll talk about solving equations. Equations are two expressions set equal to each other using an equal sign(=). When we're simplifying expressions, our end goal is to have no operations left to do. If you need more help understanding the difference between an expression and equation, you can watch this video:
When we're solving equations, our end goal is to find out what the variable (or letter) is equal to by getting the variable by itself on one side of the equal sign and a number by itself on the other side. We're going to accomplish this goal using two important steps:
Sound complicated? We'll break it down to make it easier. Let's look at an example:
5x - 4x - 6 = 18
We can start solving the same way we would start simplifying an expression, by checking the order of operations. We want to simplify each side of the equal sign as much as possible first. Looking at our equation, there are no parentheses or exponents and there's nothing to multiply or divide, so we'll just start adding and subtracting. The first part is simple: 5x - 4x is 1x, or just x.
Now we are left with this equation:
x - 6 = 18
We can't subtract 6 from x because they're not like terms (our lesson on reading algebraic expressions explains this in more detail). But x - 6 = 18 still isn't simplified enough. After all, we're looking for the value of x, not the value of x - 6.
To solve this equation, we'll need to get the x alone on one side of the equals sign. To move the -6 to the other side of the equal sign, we can use the inverse—or opposite—of -6. That would be 6. In other words, we can add six to both sides of the equation.
|+ 6||+ 6|
On the left side of the equation, -6 plus 6 is 0, and x - 0 is x. On the right, 18 plus 6 is 24, so x = 24. Now our equation is simplified. We simplified it by using the inverse of what we wanted to get rid of.
This is also called cancelling out because it lets you cancel—or get rid of—parts of an equation. This doesn't mean you can just cross out any part of the equation you don't want to solve (although that would make algebra much easier!). There are a few rules you have to follow.
First, did you notice that we added 6 to both sides of our equation? This is because the two sides of an equation must always be equal—after all, that's what the equals sign means. Any time you do something extra to one side of an equation, you have to do the same thing to the other. Because we added 6 to the -6 on the left side, we also had to add it to the 18 on the right.
|+ 6||+ 6|
Second, remember how we added six where the original expression said to subtract? We did this because 6 is the opposite of -6. To cancel out part of an expression, you'll need to use its opposite, or inverse. The opposite of subtraction is addition—and as you might guess, the opposite of addition is subtraction.
What about multiplication and division? These are opposites as well, and you can also cancel them out. For instance, how would you get the a in this equation alone on the left side of the equals sign?
5a = 30
Because the a is being multiplied by 5, you can divide both sides of the problem by 5. 5a divided by 5 is a and 30 divided by 5 is 6, so the simplified version of this equation would look like this:
a = 6