Chapter 1 Linear Equations and Linear Functions

1.4 Meaning of Slope for Equations, Graphs, and Tables

Example: Finding the Slope of a Line Find the slope of the line y = 2x + 1.

Solution We use x = 0, 1, 2, 3 in the table to list the solutions, and sketch the graph of the equation. If the run is 1, the rise is 2. So the slope is

Finding the Slope from a Linear Equation of the Form y = mx + b For a linear equation of the form y = mx + b, m is the slope of the line.

Example: Identifying Parallel or Perpendicular Lines Are the lines and 12y – 10x = 5 parallel, perpendicular, or neither?

Solution For the line the slope is For 12y – 10x = 5, the slope is not –10. To find the slope, we must solve the equation for y:

Solution For the slope is the same as the slope of the line Therefore, the two lines are parallel. We use a graphing calculator to draw the lines on the same coordinate system.

Vertical Change Property For a line y = mx + b, if the run is 1, then the rise is the slope m.

Finding the y-intercept from a Linear Equation of the Form y = mx + b For a linear equation of the form y = mx + b, the y-intercept is (0, b).

Slope-intercept form Definition If an equation is of the form y = mx + b, we say it is in slope-intercept form.

Example: Using Slope to Graph a Linear Equation Sketch the graph of y = 3x – 1.

Solution Note that the y-intercept is (0, –1) and that the slope is To graph, 1. Plot the y-intercept, (0, –1). 2. From (0, –1), look 1 unit to the right and 3 units up to plot a second point. (see the next slide) 3. Sketch the line that contains these two points. (see the next slide)

Solution

Using Slope to Graph a Linear Equation of the Form y = mx + b To sketch the graph of a linear equation of the form y = mx + b. 1. Plot the y-intercept (0, b). 2. Use to plot a second point. 3. Sketch the line that passes through the two plotted points.

Using Slope to Graph a Linear Equation Warning Before we can use the y-intercept and the slope to graph a linear equation, we must solve for y to put the equation into the form y = mx + b.

Example: Working with a General Linear Equation 1. Determine the slope and the y-intercept of the graph of ax + by = c, where a, b, and c are constants and b is nonzero. 2. Find the slope and the y-intercept of the graph of 3x + 7y = 5.

Solution 1. The slope is and the y-intercept is

Solution 2. We substitute 3 for a, 7 for b, and 5 for c in our results from Problem 1 to find that the slope is and the y-intercept is

Slope Addition Property For a linear equation of the form y = mx + b, if the value of the independent variable increases by 1, then the value of the dependent variable changes by the slope m.

Example: Identifying Possible Linear Equations Four sets of points are shown on the next slide. For each set, decide whether there is a line that passes through every point. If so, find the slope of that line. If not, decide whether there is a line that comes close to every point.

Example: Identifying Possible Linear Equations

Solution 1. For set 1, when the value of x increases by 1, the value of y changes by –3. So, a line with slope –3 passes through every point. 2. For set 2, when the value of x increases by 1, the value of y changes by 5. Therefore, a line with slope 5 passes through every point.

Solution 3. For set 3, when the value of x increases by 1, the y value does not change. Further, a line does not come close to every point, because the value of y changes by such different amounts each time the value of x increases by 1. 4. For set 4, when the value of x increases by 1, the value of y changes by 0. Therefore, a line with slope 0 (the horizontal line y = 8) passes through every point.