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
subtraction problems. However, you will also run across problems involving
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:
Think to yourself, “80 divided by what = 10?” or “What times 10 = 80?” The answer:
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