Solving Equations with Variables
The next step to solving equations is to remove the question marks, and use the letter x instead. So, rather than having a blank with a question mark on top of it, there’s going to be an x sitting there. It still means the same thing—that we want to solve for that blank—but writing it as an x is a little more advanced. The new problems would look like this:
You’re going to think about it the same way you did when there was a blank and a question mark, but instead of a question mark, there is an x there now. You would still think, 8 + what number? = 10? Or, on the other hand, 10 – 8 = what number? Either way, you find your answer to be 2.
Here’s another one with an x in it:
This is another one where you can either use your subtraction facts to think, 12 – 9 = 3, or where you can add 9 + 3, which equals 12.
Here are a few for you to try on your own, solving for x.
Think to yourself, "What plus 4 = 12?" The answer: 8.
Think to yourself, "5 minus what = 1?" The answer: 4.
Think to yourself, "What plus 7 = 14?" The answer: 7.
Think to yourself, "20 minus what = 9?" The answer: 11.
Think to yourself, "13 plus what = 18?" The answer: 5.
Solving Equations with Multiplication and Division
So far, we have only looked at equations with addition and subtraction problems. However, you will also run across problems involving multiplication and division. Here’s an example of what multiplication and a variable would look like:
In this problem, we would have to figure out what number times 5 gives us 25. We think of our 5 times tables and realize that 5 x 5 = 25, so x = 5. We can also take 25 divided by 5, and get 5 as well.
Notice also in this problem that we used a dot (·) to tell you to multiply two numbers together. This is often used instead of an x, especially when we have variables that are being multiplied together.
You will also see problems with division and variables, like this:
The easiest (and probably most common) way to think about these is to think of the times tables. What times 12 gives you 144? Well, we know that 12 x 12 = 144, so the answer must be 12. On the other hand, you could also switch the x with the given answer (12) in the problem, so you can take 144 / 12, and get 12. Either way works, so you can use whichever makes more sense.
Here are a few practice problems:
Think to yourself, "7 times what = 42?" The answer: 6.
Think to yourself, "99 divided by what = 9?" or "What times 9 = 99?" The answer: 11.
Think to yourself, "80 divided by what = 10?" or "What times 10 = 80?" The answer: 8.
Think to yourself, "9 times what = 54?" The answer: 6.
Think to yourself, "7 times what = 56?" The answer: 8.
Other times you will be given multiplication in the form of coefficients of x. A coefficient is a number that sits in front of a variable, like this: 3x. It means that you’re going to multiply 3 times x, or that there are 3 x's. You may be given an equation that looks like this: 3x = 12. In this case, you’re going to divide each “side” of the equation by the coefficient of x (3, in our example). When you divide the left side, the 3's cancel out, and all that is left is x. When you divide the right side, you get 12/3, which is 4. Thus, you’re left with x = 4. The work for this equation would look like this:
In problems like these, it is said that you will “do the opposite” of the function being performed. So, if there is multiplication, you do division; if there is addition, you do subtraction, and so on.
There may be more than one operation being performed. For example, the equation could look like this: 2x + 5 = 15. Ultimately, your goal is to get the x by itself on one side of the equation. In order to do this, we have to now eliminate 2 other numbers, the 5 and the 2. We’re going to work backwards, and get rid of the +5 first. We always get rid of numbers without variables before we get rid of coefficients. So, we look at the operation being performed, and see that it is addition (+5). We know that to get rid of addition, we have to do subtraction, so we are going to subtract 5 from both sides, like this:
After doing the first subtraction, 5 – 5 = 0, we can eliminate the 0 from our equation, so our new equation (written at the bottom of the example) is 2x = 10. Now, we look again to see what operation is being performed. We see the 3x, which we know means 3 times x, so we have multiplication. To get rid of the multiplication, we need to do division, and divide by the coefficient. Therefore, we’re dividing each side by 2, like this:
Once we divide by 2, the 2's on the left cancel out so that you just have x, and on the right, 10/2 = 5, so you’re left with x = 5.
Let’s try one more like this. We’ll give you the equation, so that you can see if you can solve it on your own. Then, you can type your answer in the box to check your answer with ours.
4x + 8 = 24
Our first step was to subtract the 8 from each side, leaving us with the equation 4x = 16. Then, we see that we have multiplication, so we have to perform division, dividing each side by 4. 16/4 = 4, so x = 4.
Solving Equations with More than One Variable
Sometimes you will be given more than one variable and asked to solve the equation. In this case, you will typically be given a numerical value for each variable, and you will have to get the answer. For example, you may be given an equation that says 2x + y = ? and then you’re given the information that x = 2 and y = 3. Then, you would simply put 2 in for x, and 3 in for y, so your new equation would look like this: 2(2) + 3 = ? Once you get this far, you just have to do the multiplication and addition (or whatever operations you have in your equation) and get an answer. For this equation, you would have to multiply 2 x 2 first, and then add 3, which is (2 x 2) + 3, which equals 7.
Let’s practice these. We’ll give you the equation and variables, and you give us the answer. Once you think you have it, type it into the box to check it.
4x + 8y = ? x = 5, y = 7
4(5) + 8(7) = ?
20 + 56 = 76
9a +6b = ? a = 3, b = 8
9(3) + 6(8) = ?
27 + 48 = 75