Algebra II: Equations of Lines

Equations involving one or two variables can be graphed on any x-y coordinate plane. In general, it is true that

if a point lies on the graph of an equation, then its coordinates make the equation a true statement, and
if the coordinates of a point make an equation a true statement, then the point lies on the graph of the equation.

The graphs of linear equations are always lines. All linear equations can be written in the form Ax + By = C, where A, B, and C are real numbers and A and B are not both zero. Futhermore, to be in standard form, A has to be a positive number. Below are examples of linear equations and their respective A, B, and C values.

Following are terms you should be familiar with:

Standard form. The form Ax + By = C for the equation of a line is known as the standard form for the equation of a line.
x-intercept . The x-intercept of a graph is the point at which the graph will intersect the x-axis. It will always have a y-coordinate of zero. A horizontal line that is not the x-axis will have no x-intercept.
y-intercept . The y-intercept of a graph is the point at which the graph will intersect the y-axis. It will always have an x-coordinate of zero. A vertical line that is not the y-axis will have no y-intercept.

One way to graph a linear equation is to find solutions by giving a value to one variable and solving the resulting equation for the other variable. A minimum of two points is necessary to graph a linear equation.

Example 1: Draw the graph of 2 x + 3 y = 12 by finding two random points.

To do this, select a value for one variable; then substitute this into the equation and solve for the other variable. Do this a second time with new values to get a second point.

Let x = 2; then find y.

Therefore, the ordered pair (2, ) belongs on the graph.

Let y = 6; then find x.

Therefore, the ordered pair (–3,6) belongs on the graph.

As shown in Figure 1 , graph these points and then connect them to make the line that represents the graph of 2 x + 3 y = 12.

Figure 1

2 x +3 y = 12.

Example 2: Draw the graph of 2 x + 3 y = 12 by finding the x-intercept and the y-intercept.

The x-intercept has a y-coordinate of zero. Substituting zero for y, the resulting equation is

Now, solving for x,

The x-intercept is at (6,0).

The y-intercept has an x-coordinate of zero. Substituting zero for x, the resulting equation is

Now, solving for y,

The y-intercept is at (0,4).

The line can now be graphed, as shown in Figure 2 , by plotting these two points and drawing the line they determine.

Figure 2

x- and y- intercepts.

Notice that Figures 1 and 2 are exactly the same. Both are the graph of the line 2 x + 3 y = 12.

Example 3: Draw the graph of x = 2.

As shown in Figure 3 , x = 2 is a vertical line whose x-coordinate is always 2.

Figure 3

x = 2 for all y values.

Example 4: Draw the graph of y = −1.

As shown in Figure 4 , y = –1 is a horizontal line whose y-coordinate is always –1.

Figure 4

y = –1 for all x values.

Suppose that A is a particular point called ( x₁, y₁) and B is any point called ( x, y). Then the slope of the line through A and B is represented by

Apply the cross products property, and the equation becomes

This is the point-slope form of a nonvertical line.

Example 5: Find the equation of the line containing the points (–3, 4) and (7, 2) and write the equation in both point-slope form and standard form.

For the point-slope form, first find the slope, m.

Now, choose either given point, say (–3, 4), and substitute the x and y values into the point-slope form.

For the standard form, begin with the point-slope form and clear it of fractions by multiplying both sides by the least common denominator.

Multiply both sides by 5.

Get x and y on one side and the constants on the other side by adding x to both sides and adding 20 to both sides. Make sure A is a positive number.

A nonvertical line written in standard form is Ax + By = C, with B ≠ 0. Solve this equation for y.

The value becomes the slope of the line, and becomes the y-intercept value. If is replaced with m and is replaced with b, the equation becomes y = mx + b. This is known as the slope-intercept form of a nonvertical line.