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Linear Equations

Definitions

A linear equation in two variables has the form y = mx + b for some constants m and b. The graph is a line with slope m and y-intercept (0,b). It is defined for all real numbers x since any real number can be substituted for x .  For any non-horizontal line, the y values also take on all real numbers.

Any linear equation has a constant rate of increase or decrease.  This constant rate of change is the slope of the line and is represented by m in the equation. Use the following interactive example to explore how the graph of the line y = mx + b  changes as the values of the coefficients m and b change. Note any useful observations in your journal.

Slope of a Line

The slope is the constant rate of change of a linear equation. It can be thought of as the ratio of the vertical change to the horizontal change between two points on the graph of a line.  If the two points are (x1, y1) and (x2, y2), then the vertical change is y2 - y1 and the horizontal change is x2 - x1.  Hence we can use the formula

                                   

to determine the slope of a line if we know two points on that line.  See Figure 1 below.

 

Explore the following interactive example to see how the slope of a line is computed when two points are known. What observations can you make?

In general, when the slope of a line is positive, the line will be increasing. That is, the graph will be a line that rises from left to right. When the slope is negative, the line will be decreasing and the graph will be a line that falls from left to right. If the slope of a line is 0, then it is neither increasing nor decreasing, but is constant. The graph of a constant is a horizontal line.

Use the next interactive example to see how different values of the slope, m, affect the graph of the line y = mx + b.

Horizontal and Vertical Intercepts

The points where a line crosses the vertical and horizontal axes are known as the vertical and horizontal intercepts. (These points are often referred to as the x-intercept and the y-intercept.) Given a linear functiony = mx + b,

  1. The vertical intercept (y-intercept) is found by evaluating the equation when the input variable, x, is 0 and is always the same as the constant b.

The horizontal intercept  (x-intercept) is the value of the variable x when the y value is 0. It is found by solving the equation 0 = mx + b.

Interactive Example

Now explore how the values of the y-intercept, b, affect the graph of the line y = mx + b.

Algebraic Example

Find the vertical and horizontal intercept of the line y = -7.2x + 250.

Solution

 When x = 0, y = -7.2(0) + 250 = 250, so the vertical intercept is 250. This means that the graph crosses the horizontal axis at the point (0, 250). Also notice that this is the value of b in the general form y = mx + b.

To find the horizontal intercept we can replace y with 0 and solve the linear equation 0 = -7.2x + 250.  The solution is given below. 

0 = -7.2x + 250

-250 = -7.2x

34.7 = x

The horizontal intercept is 34.7. This is the point (34.7, 0) on the graph of the line. A graph of the line is shown in Figure 2.

            Figure 2