Solve for a Variable

Algebra is when, in math, you use variables in place of numbers that you don't know.  Those variables stand for a number, you just don't know what it is.

When you are given an algebraic equation with just one variable, you have all of the information you need to figure out the value of that variable.

Think about this example:

$x + 5 = 9$

Do you know what $x$ equals?

Sure you do.  You know that $4+5=9$ so $x$ must equal $4$.

That's algebra.  But, often we use algebra in much more complicated equations, so you use certain steps to figure out the value of the variable. 

How do you do it? Essentially, what you want to do is isolate the variable. When you see a variable mixed in with lots of other numbers, it's hard to see what it equals, so you want to get it alone.  You want to rearrange your equation so that the variable is alone on one side of the equal sign, and you have some combination of numbers on the other side.  When you do the math with those numbers, you will have a value for your variable. 

How do you isolate a variable? The rule in equations is that, as long as you keep the equation in balance, you can perform mathematical operations to isolate your variable.

What does that mean?  It means that you can do math to your equation, as long as, whatever you do to one side of the equation (to the terms on one side of the equal sign), you also do to the other side of the equation (to the terms on the other side of the equal sign).

What math do you want to do?  You are trying to isolate your variable.  So, take the numbers that are attached to your variable, and "undo" them.

How do you "undo" a number? You do the opposite operation (also called the inverse operation).  So, if there is a $+3$ attached to your variable, you want to subtract 3. If your number is multiplied by 5, you want to divide by 5.  And remember, whatever you do to one side the equation, you also have to do to the other (keeping the equation in balance is what makes any operation you want to do mathematically "legal"). Here's a table showing opposite or inverse operations:

Operation          Opposite or Inverse Operation

$+$                        $-$

$-$                         $+$

$\times$                       $\div$

$\div$                         $\times$

$x^2$                      $\sqrt{x^2}$

(Note:  The opposite of a square is a square root, the opposite of a cube is a cube root, etc.)

$\sqrt{5}$                 $\sqrt{5}^2$ 

(Note: The opposite of a square is a square root, the opposite of a cube is a cube root, etc).

Do you have to "undo" the numbers in any particular order? You don't have to.  But, it's usually easiest if you undo numbers in reverse order of operations. Order of operations says that you do addition and subtraction last, so undo addition and subtraction first.

Let's try an example:

$$\eqalign{2x + 5 &= 12 &&\text{We want to isolate the x.}\\2x\mathbf{+5}&=12 &&\text{Remove addition and subtraction first}\\-5 & \quad -5 &&\text{The opposite of +5 is -5.  Subtract 5 from both sides}\\2x&=7&&\text{Rewrite what's left of your equation}\\\mathbf{2}x&=7&&\text{Now you need to get rid of the 2, which is attached with multiplication}\\\div 2 & \quad \div 2&&\text{Division is the opposite of multiplication.  Divide each side by 2}\\x&=3.5&&\text{You have isolated your variable.  x equals 3.5}}$$

Let's try an example with a power in it:

$$\eqalign{(2x)^2+4&=8\\-4 & \quad -4\\(2x)^2&=4\\\sqrt{(2x)^2}&=\sqrt{4}\\2x&=2\\x&=1}$$