Algebra Topics -
Solving Equations
Algebra Topics
Solving Equations
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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.
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.
| x | - 6 | = | 18 |
| + 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.
| x | - 6 | = | 18 |
| + 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.
Watch the video below to see this example problem solved.
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
Watch the video below to see this example problem solved.
Let's look at another example:
4(2x + 3) = 68
First, we need to look to see if anything can be simplified. Remember in the previous section we talked about the number on the outside of the parentheses meaning multiplication? According to that, we can multiply 4 · 2x and 4 · 3. 4 · 2x is 8x and 4 · 3 is 12.
8x + 12 = 68
This gives us 8x + 12 = 68.
Now that both sides of the equal sign are simplified, we will need to use canceling to get x by itself. Right now we have two things we need to move, the 8 and the 12. We're adding 12, so we would subtract to move it. We're also multiplying x by 8, so we would divide to move it. But which one do we move first?
Remember, canceling uses inverse - or opposite - operations. Since we're using opposite operations to move things, we're going to use the opposite of the order of operations to decide which order to move them in.
The order of operations says we would simplify multiplication and division before addition and subtraction, so we're going to do the opposite. We're going to use addition/subtraction first, and then multiplication/division.
First, we'll subtract 12 from both sides:
| 8x | +12 | = | 68 |
| - 12 | -12 |
Since 12 - 12 is 0, we're left with 8x on the left. Since 68 - 12 is 56, we're left with 56 on the right.
| 8x | = | 56 |
| / 8 | / 8 |
Finally, we'll divide. 56/8 = 7
x=7
We're done! This means for 4(2x + 3) = 68, x has to equal 7.
Watch the video below to see this example problem solved.
Let's practice what you just learned by going through a few more problems. Remember, in order to simplify these we'll use the order of operations and cancelling out.
Pay attention to the steps we take to simplify these expressions—in a bit, you'll have a chance to solve a few on your own.
Simplify this expression to find the value of x:
6x + 23 = 74
Take a moment to think about what you'd do first. You might even want to get out a piece of paper to see how you would simplify this on your own. Once you're ready, keep reading to see how we got the correct answer.
Just like we did on the last page, we'll start by seeing if there's anything we can do with the order of operations. This expression has two operations: addition and an exponent.
6x + 23 = 74
According to the order of operations, we need to calculate the exponent first. That's 23, which is equal to 2 ⋅ 2 ⋅ 2, or 8.
6x + 23 = 74
The order of operations says we should add next, but we can't add 6x + 8—a variable with a coefficient like 6x can only be added to another like term. (In other words, a number with the variable x can only be added to another number with the variable x.) In order to get 6x on its own, we'll have to cancel out + 8.
6x + 8 = 74
We can do that with the opposite of 8, which is - 8. We'll subtract 8 from both sides of the equals sign. 8 - 8 is 0. 74 - 8 is 66.
| 6x | + 8 | = | 74 |
| - 8 | - 8 |
We're almost done. All that's left to do is get rid of the 6 in 6x. Remember, 6x is just another way of writing 6 ⋅ x.
6x = 66
Because 6 and x are being multiplied, we can cancel out the 6 by doing the opposite: dividing.
| 6x | = | 66 |
| 6 | 6 |
6x / 6 is x and 66 / 6 is 11, so x = 11. We're done!
x = 11
As you might have noticed, you don't have to follow the order of operations once you start cancelling out. All that matters is keeping both sides of the expression equal. In fact, it's best to cancel out addition and subtraction first.
Let's try another problem. Simplify for y.
4 (3y - 8) = 4
This problem is slightly different from the last one, but it uses the same skills. Here's how to solve it:
According to the order of operations, we'll need to simplify the expression in parentheses first. However, we can't subtract 8 from 3y—we can't subtract a number from a variable.
4 (3y - 8) = 4
Because 4 is next to the parentheses, we're supposed to multiply what's in the parentheses by 4. (Confused? Review our lesson on reading algebraic expressions).
4 (3y -8) = 4
4 ⋅ 3y is 12y and 4 ⋅ -8 is -32. You can't subtract 32 from 12y, either, so to simplify this expression any further we'll have to start canceling things out.
12y - 32 = 4
Let's get rid of the -32 first. The opposite of -32 is 32, so we'll add 32 to both sides. -32 + 32 is 0, and 4 + 32 is 36.
| 12y | - 32 | = | 4 |
| + 32 | + 32 |
We're almost finished. We just have to cancel out of the 12 in 12y. Remember, 12y could also be written as 12 ⋅ y.
12y = 36
Because 12 and y are being multiplied, we can cancel out the 12 by dividing.
| 12y | = | 36 |
| 12 | 12 |
12y / 12 is y, and 36 / 12 is 3. We did it: y equals 3.
y = 3
Try solving the next few problems on your own. The answers are below.
Simplify this expression to find the value of x:
-2 + x / 5 - 3 = 0
Find the the value of y:
3 (y + 2y) = 36
Find the the value of r:
300r - 60r + 102 = -380
Believe it or not, you now have the tools to simplify many expressions, even complicated-looking ones like this:
3x - 24 ⋅ 2 = 8x + 2
This might look more difficult than the problems you solved on the last page, but you'll use the exact same skills to solve this one. The major difference between this expression and the others you solved is that this one has a variable and at least one number on both sides of the equals sign—so you'll have to do a bit more cancelling out.
You'll also have to choose whether you want the variable on the left or the right side of the equals sign in your simplified expression. It doesn't really matter—the answer will be the same either way—but depending on the problem, you might find that the math feels easier one way than another. No matter what, though, your simplified equation should have only a variable on one side of the equation and only a number on the other.
Let's try the problem from the top of the page: 3x - 24 ⋅ 2 = 8x + 2.
First, we'll want to handle what we can with the order of operations. It looks like all we can do is multiply -24 ⋅ 2. Everything else involves adding or subtracting unlike terms: -24 ⋅ 2 is -48.
3x -24 ⋅ 2 = 8x + 2
Let's try to get x on the left side of the equals sign and the number on the right. We'll start by cancelling out -48 on the left. We can do this by adding 48 to both sides. -48 + 48 is 0, and 2 + 48 is 50.
| 3x | -48 | = | 8x | + 2 |
| + 48 | + 48 |
Because we decided that x will be on the left side, we have to get rid of 8x on the right. We can do this by subtracting 8x from both sides. 8x - 8x is 0, and 3x - 8x is -5x.
| 3x | = | 8x | + 50 |
| - 8x | - 8x |
Now all that's left to do is to get rid of the -5 in -5x. Because -5x is a way of writing -5 ⋅ x, we can cancel it by dividing both sides by -5. -5x / -5 is x, and 50 / -5 is 10.
| -5x | = | 50 |
| -5 | -5 |
We're done! x is equal to -10.
x = -10
As you can see, simplifying this equation really wasn't much more complicated than simplifying any of the other equations in this lesson—it just took a little longer.
Watch the video below to see this example problem solved.
Now it's your turn. Try simplifying these longer expressions.
Solve for i.
-46 -2i = 42 + 7i ⋅ 6
Solve for j.
90j / 5 + 22 = 140 + j
Solve for k. (Hint: your final answer will be a fraction.)
3 + (3k + 6k) = 3k + 5
Sometimes you might see an equation with more than one variable, like this one:
2x + 6y -10 = 38
If an expression has more than one variable, you won't be able to simplify it all the way—there's not enough information. Instead, problems with equations that have multiple variables will usually ask you to solve for one of the variables. You'll simplify it as much as you can, with the variable you're solving for on one side of the equation and any other numbers and variables on the other. Let's simplify the expression above: 2x +6y - 10 = 38.
We can't do anything with the order of operations, so let's start cancelling things out. We want x alone on the left side, so we'll try to get everything else on the right.
2x + 6y - 10 = 38
First, we'll cancel out -10. The opposite of -10 is 10, so we'll add 10 to both sides.-10 + 10 is 0, and 38 + 10 is 48.
| 2x | + 6y | - 10 | = | 38 |
| + 10 | + 10 |
Next, let's get rid of 6y. We'll subtract it from both sides. 6y - 6y is 0. Because there's nothing to subtract it from on the other side, we'll just write -6y on the right. (Confused? It's like we subtracted 6y from nothing, or 0—and 0 - 6y is -6y.)
| 2x | + 6y | = | 48 | - 6y |
| - 6y |
Now we have to get rid of the 2 in 2x. Because 2x is another way of saying 2 ⋅ x, we'll divide both sides by 2 to get x alone on the left. 2x / 2 is x, and (48 - 6y) / 2 is 24- 3y.
| 2x | = | 48 | - 6y |
| 2 | 2 | ||
That's all it takes! The expression isn't fully simplified—we still don't know the numerical value of x and y—but it's simplified enough because we can say that x equals 24 - 3y.
x = 24 - 3y
Remember, your goal with problems like this isn't to completely simplify the expression—it's to find the value of one of the variables.
It is actually possible to solve for two variables when you have more than one equation with the same variables. This is called a system of equations. We actually use systems of equations in our lesson on distance word problems, but we don't discuss how they work in general. To learn more about systems of equations, check out this video from Khan Academy.
Watch the video below to see this example problem solved.
Solve for r.
88q + 4r - 3 = 5
Solve for s. (Hint: your final answer will be a fraction with a denominator of r.)
(13sr) / 2 = 39
Solve for m.
6m - 30p / 5 = 12
It's important to check your work in algebra, especially when you're first getting started. Luckily, checking your work when you're simplifying equations is pretty straightforward. All you have to do is replace the variable in the equation with the value you found when you simplified it. To see how this works, let's look back at one of the equations we simplified before:
4 (3y - 8) = 4
We found that y was equal to 3. Let's see if we got the answer right.
Here's our original equation. y is our variable, so we'll be replacing it with the value we found: 3.
4 (3y - 8) = 4
Here's what the equation looks like with 3 instead of y. Now we're going to see if the equation is true. If the left side is equal to the right side, our answer is correct.
4 (3 ⋅ 3 - 8) = 4
We'll follow the order of operations, with parentheses first. 3 ⋅ 3 is 9, and 9 - 8 is 1.
4 (1) = 4
Now that we've simplified the parentheses, all we have to do is multiply 4 times 1.
4 (1) = 4
4 ⋅ 1 is 4. Both sides of our equation are equal, so our answer is correct!
4 = 4
That's all it takes! Checking every expression you simplify is a good habit to get into, and you'll find that checking your work usually takes less time than it took to simplify the equation in the first place.
Let's try one more:
The expression we'll be looking at is 5x + 3 = 23 + x. We're checking to see if the solution x = 4 is correct.
5x + 3 = 23 + x
First, we'll replace the variable x with 4.
5 ⋅ 4 + 3 = 23 + 4
To check our work, we'll have to simplify both sides of the expression. We'll start with the left side. According to the order of operations, we need to multiply first, then add. 5 ⋅ 4 is 20, and when you add 3 to that, you get 23.
5 ⋅ 4 + 3 = 23 + 4
Now we need to simplify the right side: 23 + 4 is 27.
23 = 23 + 4
Our equation can't be right—23 and 27 are not equal. We now know that x does not equal 4. In other words, the answer is incorrect.
23 = 27
As you just saw, if you're checking a problem and the final expression is not a balanced equation, your answer is not correct. Take time to go back and simplify your original equation again. On your second try, pay careful attention to the order of operations, and make sure you're adding, subtracting, multiplying, and dividing correctly.
Want to check that last problem again? This time, check it with x = 5.
Check this problem. Is u = 6 the correct answer? If not, what is?
u (3 + 8) / 2 = 33
Check this problem. Is v = 5 the correct answer? If not, what is?
v / 5 + 20v = 19v + 12
Check this problem. Is w = 8 the correct answer? If not, what is?
5w + 3 = 4w + 10
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