The key step is to multiply both sides by x to extract the variable from the denominator. It is not at all uncommon to have to move the variable from side to side in order to isolate it.

 

    Remember, performing an operation on a variable is mathematically no different than performing that operation on a constant or any other quantity.

 

    Here’s another, slightly more complicated example:

 

    This question is a good example of how it’s not always simple to isolate a variable. (Don’t worry about the logarithm in this problem—we’ll review these later on in the chapter.) However, as you can see, even the thorniest problems can be solved systematically—as long as you have the right tools. In the next section, we’ll discuss factoring and distributing, two techniques that were used in this example.

 

    So, having just given you a very basic introduction to solving equations, we’ll reemphasize two things:

1. Do the same thing to both sides.
2. Work backward (with respect to the order of operations).

    Now we get into some more interesting tools you will need to solve certain equations.

 

 

    Distributing and factoring are two of the most important techniques in algebra. They give you ways of manipulating expressions without changing the expression’s value. So it follows that you can factor or distribute one side of the equation without doing the same for the other side of the equation.

 

    The basis for both techniques is the following property, called the distributive property:

 

    Similarly:

 

    a can be any kind of term, from a variable to a constant to a combination of the two.

 

 

    When you distribute a factor into an expression within parentheses, you simply multiply each term inside the parentheses by the factor outside the parentheses. For example, consider the expression 3y(y² – 6):

 

    If we set the original, undistributed expression equal to another expression, you can see why distributing facilitates the solving of some equations. Solving 3y (y² – 6) = 3y³ + 36 looks quite difficult. But if you distribute the 3y, you get:

 

    Subtracting 3y³ from both sides gives us:

 

 

    The expression simplifies further:

 

    See how useful these techniques are? You can group or ungroup quantities in an equation to make your calculations easier. In the last example from the previous section on manipulating equations, we distributed and factored to solve an equation. First, we distributed the quantity log 3 into the sum of x and 2 (on the right side of the equation). We later factored the term x out of the expression x log 2 – x log 3 (on the left side of the equation).