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Solving One-Step Equations

Finally we are getting into the kinds of problems that you usually think of when you imagine algebra, the ones where you solve for x.

There is one extremely important rule to follow when solving all algebraic problems:

Whatever you do to one side of an equation, you MUST do to the other.

Equations are like carefully balanced scales. Imagine an old-timey scale. If both expressions on each side of the equal sign match, then they are balanced.

balanced scale

If one side is heavier than the other, the scales are tipped.

scale 1

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: x + 4 = 6. 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.

scale 3

In order to solve for x, we must isolate (get all by its lonesome self) the variable. To do this we must get rid of that pesky 4. If we subtract 4 from the left side, the scale will be unbalanced.

scale 4

To counter this, we must also subtract 4 from the other side of the equation.

scale 5

Now the scales are balanced once again and all is right with the universe.

x + 4 = 6

x + 4 -4 = 6 -4

x = 2

scale 6

Don't worry, we really don't expect you to draw scales each time you need to solve an equation. We are just using this to illustrate a very important point, you must keep algebraic equations balanced at all times. In order to do this, whatever you do to one side of the equation must be done to the other.

How to Isolate the Variable

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 (subtraction is the opposite of addition). Here are some ways to undo other operations:


OperationHow to UndoExample
AdditionSubtraction

Or

Add the Opposite

2 + -2=0

x + 1 = 5subtract 1
x + 1 - 1 = 5 - 1
x = 4
SubtractionAddition
x - (-2) = -7add (-2)
x - (-2) + (-2) = -7 + (-2)
x = -9
MultiplicationDivision
3x=21divide by 3
3x/3=21/3
x = 7
DivisionMultiplication
x/-2=10multiply by (-2)
(x/-2)x -2 = 10 x -2
x = -20

How to Check Your Answer

One of the best things about solving equations is that you can, and should, plug your answer back into the original equation to see if it works. Let's look at that last division problem.

x/-2 = 10

(x/-2)(-2) = 10(-2)

x = -20

If we plug -20 back into the original equation, both sides should be equal.

-20/-2 = 10

10=10

Since both sides equal 10, we know that our answer is correct!

Look Out: be sure that you are solving for the variable, not the opposite of the variable (-x).

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