Finally, we're getting into the kinds of problems that most people usually think of when they imagine algebra: the ones where we solve for x.
There's one extremely important rule to follow when solving all algebraic problems:
Equations are like carefully balanced scales. Imagine an old-timey scale. If both expressions on each side of the equal sign match, then they're balanced.
If one side is heavier than the other, the scales are tipped.
In algebra, we solve equations for the missing variable. The trick is to keep the scales balanced during all steps.
Let's start by looking at a simple example:
We know that you know that we know you know the answer (2), but for argument's sake, let's use our scales to solve this.
In order to solve for x, we must isolate the variable, or get it all by its lonesome self. To do this, we'd better get rid of that pesky 4. If we only subtract 4 from the left side, the scale will be unbalanced.
To counter this, we must also subtract 4 from the right-hand side of the equation.
Now the scales are balanced once again, and all is right with the universe.
Don't worry; we really don't expect you to draw scales each time you need to solve an equation. We're just using this to illustrate a very important point: we must keep algebraic equations balanced at all times. In order to do this, whatever we do to one side of the equation must be done to the other.
The most straightforward way to get a variable alone is to undo the operation that accompanies it. In the equation above, 4 is added to x. To undo this, we subtract 4 because subtraction is the opposite of addition. Here are some ways to undo other operations:
|Operation||How to Undo||Example|
Add the Opposite
|x + 1 = 5|
(Subtract 1 from both sides.)
x + 1 – 1 = 5 – 1
x = 4
|Subtraction||Addition||x – 2 = -7 |
(Add 2 to both sides.)
x – 2 + 2 = -7 + 2
x = -5
(Divide both sides by 3.)
(Multiply both sides by -2.)
One of the best things about solving equations is that we can, and should, plug our answer back into the original equation to see if it works. Let's look at that last division problem.
If we plug -20 back into the original equation, both sides should be equal.
Since both sides equal 10, we know that our answer is correct!
Look Out: be sure that you're solving for the variable, not the opposite of the variable (-x).