Fascinating Linear Equations


Let’s practice solving algebra (linear) problems. It seems really difficult at first, but I promise you, after some practice, it will get easier. The thing I love about math is that you have a bunch of rules and all you have to do is apply them (memorize less than say History!).

Let’s solve for x in the following problems. At this point, don’t worry about where we got the variables and numbers – let’s just go through the mechanics. We’ll do real algebra word problems later.

Again, you want to get the x by itself – when it’s by itself, we get the answer of what x is! To do this, you have to move things around by doing the opposite of what you have. For example, if the problem has addition in it, you have to subtract from both sides; if the problem has division, you have to multiply on both sides. Remember also that the sign of a variable or constant is what is in front of it; sometimes it’s an “invisible +” if it’s at the beginning.

Step 1

First combine the numbers (called “constants”) and the same letters (variables) on each side if you need to. This is called combining like terms, meaning you can add the same letters together – in this case the x’s. As we saw earlier, this is because of the Distributive Property, and later we’ll see we can do the same for x2, or actually any term with identical variables. Remember that when we just have an x, it’s the same as 1x. We can change the order of our terms using the Commutative Property of Addition. So let’s say we are trying to solve 4x−4−x=4+1 for x:

Step 2

Add or subtract the “numbers” (“constants”) from each side to get rid of them on the side where the variable (the x) is. You want to do the opposite or (additive inverse) of what we have to get rid of. If you are subtracting a number, you want to add on both sides, and if you are adding a number, you want to subtract on each side. Line up the letters and numbers vertically. In our case, we are subtracting on each side, and using the Subtraction Property of Equality:

Step 3

If there’s a number before the x (which means it’s multiplied by the x), divide both sides by that number (using the Multiplicative Inverse). If there’s a number below the x (which means it’s divided by the x), multiply both sides by that number. If there’s a fraction before the x, we need to multiply both sides of the reciprocal of the fraction. (The numbers immediately before the variables are called coefficients.) In our case, we have a number before the x, and we are using the Division Property of Equality.

Now we have solved to get what x is: x=3!

Step 4

We should always try to check our answer, by plugging the answer we got back in for wherever x is:

It doesn’t really matter what order you do the operations, as long as you do the addition and/or subtraction first, and then do the multiplication and/or division. Do you see how this is sort of the opposite of PEMDAS – it’s SADMEP! So first do the addition and subtraction, then the multiplication and division, and then worry about things in parentheses.

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