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Linear Functions and Straight Lines

**Linear
Functions: Slope-Intercept Form**** · Slope · Intercept · Point-Slope Form · ****Examples**** · Recommended
Books**

**Linear
Functions: Slope-Intercept Form**

Linear functions describe **straight lines** and have the general form

The number m is called the **slope** and determines the tilt of the line, while b is
called the **y-intercept**, the point on the y-axis where the line crosses at x = 0.
This form of a linear equation is often called **slope-intercept form**.
A typical graph of one of these functions looks like this:

The slope is calculated from knowing any two distinct points on the line, (x_{1},
y_{1}) and (x_{2}, y_{2})

If the points are distinct, then the x coordinates cannot be the same and so zeros in the denominator are avoided.

The y-intercept can be calculated from any point if the slope is known. Since for any
point (x_{1}, y_{1})

we must then have

Given any two *distinct* points on the line, (x_{0}, y_{0}) and
(x, y), we can set up two versions of the equation for the line

Subtracting, we get

This is the **point-slope form**.

**Slope-Intercept
and Point-Slope Forms from Two Points · Y-Intercept From Point-Slope Form · Intercepts
and Slope-Intercept Form · Crossing Lines · Perpendicular Lines · Parallel Lines**

Example - Slope-Intercept and Point-Slope Forms from Two Points

Find the equations in slope-intercept and point-slope form for the line that passes through the points

Solution

First calculate the slope:

Then take one of the points and calculate the intercept:

As a check, let’s do the same calculation with the other point:

We got the same result with both points, a good indication that the calculations are correct. The equation of the line in slope-intercept form is

and the equation in point-slope form is

Example - Y-Intercept From Point-Slope Form

Given the following equation of a line in point-slope form:

Find the y-intercept.

Solution

There are two ways to do this problem. The first is to multiply out the right-hand side of the equation and subtract 1 from both sides:

So we now know the intercept is –7. The other way to do this problem is to use the intercept formula:

We arrived at the same answer.

Example - Intercepts and Slope-Intercept Form

The y-intercept of a line is 3 and the x-intercept is 2. What is the equation of the line in slope-intercept form?

Solution

The y-intercept of a line is the y-coordinate where x = 0. Therefore, if the y-intercept is 3, the point on the line is (0, 3). In similar fashion, the x-intercept is the point on the x-axis where y = 0. Thus, if the x-intercept is 2, the point is (2, 0). With this information, we can apply the methods of the first example. First, calculate the slope:

Now, we don’t have to calculate the y-intercept because it was given as 3! The equation of this line is:

Example - Crossing Lines

At what point do the following two lines cross?

Solution

The lines cross at a single point, so the lines must have the same x and y coordinates at that point.

Now that we have the x-coordinate, we can use either equation to find the y-coordinate:

Now we have both coordinates, so the answer is:

Example - Perpendicular Lines

Find the line perpendicular to

that crosses this given line at (1, 17/6).

Solution

This example requires a formula for perpendicular lines, whose slopes are related by

So for our second (perpendicular) line, the slope is

We know that the point (1, 17/6) belongs to both lines, so we might as well use it. Plugging into the formula for the y-intercept,

Thus the equation for the perpendicular line is

Example - Parallel Lines

Find the line parallel to the line

which passes through the point (-5, 1).

Solution

Parallel lines have the same slope, so we already know that m = 2. Plugging in to get the intercept yields

So the final line is

College Algebra (Schaum's Outlines)

The classic algebra problem book - very light on theory, plenty of problems with full solutions, more problems with answers

Schaum's Easy Outline: College Algebra

A simplified and updated version of the classic Schaum's Outline. Not as complete as the previous book, but enough for most students