Parallel and Perpendicular Lines

Many of the problems that we will encounter in this section involve parallel or perpendicular lines.  To study this we must focus on the slopes of the lines.

[ Interactive: Parallel Lines ]

Instructional Video: Parallel and Perpendicular Lines - Part 1 

There is notation that corresponds to these ideas, for example,

reads "m perpendicular".  To find the slope of the perpendicular line simply reciprocate the number and change the sign.

[ Interactive: Perpendicular Lines ]

Find the corresponding perpendicular slope.

Of course, just use the same slope if you are asked to find the slope a parallel line.

Are the lines parallel, perpendicular or neither?

Find the slope of the line perpendicular.

Now we use these facts and the formula for equations of lines to find equations given certain geometric information. Recall,

Example: Find the equation of the line perpendicular to y = −1/4 x + 2 passing through the point (−1,−5).

      

This three step process using slope-intercept form always works. We could find this equation using point-slope form as well as demonstrated below.

Example: Find the equation of the line perpendicular to y = −1/4 x + 2 passing through the point (−1, −5).

    

Notice that the answer y = 4x − 1 is the same using either method.

Instructional Video: Parallel and Perpendicular Lines - Part 2

Find the equation of the line. (Using slope-intercept form.)

  

Find the equation of the line. (Using point-slope form.)

  

Video Examples on YouTube:

Simplifying Algebraic Expressions

The properties of real numbers are very important in our study of Algebra. These properties can be applied in Algebra because a variable is simply a letter that represents a real number. The distributive property is one that we apply often when simplifying algebraic expressions. Given real numbers a, b, c:

a(b + c) = ab + ac

When multiplying an expression within parentheses you must multiply everything inside by the number or variable that you are distributing.

Introduction to the Distributive Property

Simplify.

When simplifying, we will often have to combine like terms after we distribute.  This step is consistent with the order of operations, multiplication before addition.

Combining Like Terms

Simplify.

Simplifying Algebraic Expressions

To combine like terms, the variable parts have to be exactly the same.  But before combining like terms, generally, we will first distribute if necessary.  When distributing negative numbers notice that the operations change.

Simplify.

Profit Word Problem: Profit is equal to revenues less cost of production.  If the revenue R can be represented by

and the cost C can be represented by

where x represents the number of units produced, find an equation that represents the profit.

Subtracting Variable Expressions: What is the difference between 3x − 4 and −2x + 5?

Video Examples on YouTube:

      

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