copyright 2015, 2008, 2011 pearson education, inc. section 1.4, slide 1 chapter 1 linear equations...

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 1

Chapter 1Linear 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.


SolutionWe 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

runr 2e .

1is 2m


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, 5 36

y x

perpendicular, or neither?


Solution

For the line the slope is3,56

y x 56

.

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

12 10 5y x

1012 10 105xy x x

12 10 5y x

12 1012 1 2

52 1

y x

5 56 12

y x


Solution

For the slope is the same as the 12

56

5 ,y x 56

,

slope of the line Therefore, the two 3.56

y x

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 theslope is To graph, 3 rise3 .

1 run

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.

riserun

m


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. ax by c axax axby c by ax c by a cb b

xb

a cy xb b

The slope is and the y-intercept is ,ab

0, .cb


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 is3

7 and the y-intercept is

50, .7


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.

copyright 2015, 2008, 2011 pearson education, inc. section 1.4, slide 1 chapter 1 linear equations...

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