LINEAR EQUATIONS

Linear equations are the first type of equation most students encounter. You are expected to learn everything about them. It is essential that algebra students get really good at working with these equations because everything that comes after them builds on the concepts presented here. While they may seem pretty basic, they will continue to be used throughout your math studies for the next several years. Several calculus topics rely on your really understanding what a "slope" is and you will need to quickly find the equation of a line given various facts. Thus, it is important that you start this stage of your math education with a firm foundation.

What is a linear equation?

A linear equation is an equation involving variables where their exponents are nothing more than 1. The table below shows some examples of equations that are linear, and some that are not.

Linear	Non-Linear

If you graph a linear equation on a Cartesian plane (the xy-axes), the resulting graph is a straight line. To graph a linear equation, you need to plot the ordered pairs that satisfy the equation. For example, given the equation

You could create a table of values that make the equation "true." One such table is:

These points are then "plotted" and a straight line drawn through them, extending past the "end points" (for there really are no "end points" of this line). We could use the pair

for

and still have a "sum" of 8. Thus, the completed graph is shown below:

The points do not have to be shown as "big dots." A "good graph" would display only the line and "dots" would be used only to show important points.

Slope (gradient) of a line:

The slope of a line is the rate at which it is changing as you move from left to right along it. The slope formula is:

Many refer to slope as the "rise over run." The key to calculating slopes correctly is to put the vertical change on the top of the fraction. [The term "gradient" is often used to mean the same as "slope." If you have heard of a steep "grade" of a hill, you can now guess what this means.]

Parallel lines:
Two lines are parallel if and only if they have the same slope.
Notation: line 1 is parallel to line 2: .

Perpendicular lines:
Two lines are perpendicular if and only if the product of their slopes is -1. Thus, .
Notation: line 1 is perpendicular to line 2: .

Coincident lines:
Two lines are coincident if and only if they are of the following forms :

        ,

Meaning: the equation of the first line differs from that of the second by only a scalar multiple.

Example of coincident lines

The graphs of two coincident lines will be the exact same graph.

Common point of two lines:

The common point of two lines is the point where the two lines intersect. Two lines intersect if and only if they lie in the same plane and have different slopes. The point of intersection can be found algebraically by any of several methods: substitution, elimination, using matrices with Cramer’s rule, or by solving matrix equations (often done with graphing calculator assistance when there are more than 3 unknowns and equations). Points of intersection may also be found using your grapher (though these are often only approximations).

Slope-Intercept form of a linear equation:
m = slope (gradient), b = the y-intercept. This form is particularly useful for creating a table of values.

Point-slope form of a linear equation:
This form of a linear equation is often used when finding the equation of a line if a point on the line and the slope are known.

Horizontal lines are all of the form: y = c

Vertical lines are all of the form: x = c

Required skills:

• Graphing a line knowing a pair of points or a point and the slope.
• Finding the equation for a line given a point and the slope OR if given only two points.
• Finding x- and y-intercepts (points where the line crosses an axis).
• Finding solutions to systems of the form

  by elimination, substitution, and using the graphing calculator.

• Given a line and a point not on the line, find the equation of lines through that point that are either parallel or perpendicular to the original line.

<<< Back to Mr. Whitney's Mathematics Home Page	Site Created & Maintained by
	Matthew C. Whitney