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Meaning of Slope for Equations, Graphs, and Tables
Section 1.4
Lehmann, Intermediate Algebra, 3edSection 1.4
Find the slope of the line
Slide 2
Finding Slope from a Linear Equation
2 1.y x
x y
0 1
1 3
2 5
3 7
Create a table using x = 1, 2, 3. Then sketch the graph.
rise 22
run 1m
Example
Solution
Finding Slope from a Linear Equation
Lehmann, Intermediate Algebra, 3edSection 1.4
Note the following three observations about the slope of the line
Slide 3
Finding Slope from a Linear Equation
2 1.y x 1. The coefficient of x is 2, which is
the slope.
2. If the run is 1, then the rise is 2.
3. As the value of x increases by 1, the value of y increases by 2.
Observations
Finding Slope from a Linear Equation
Lehmann, Intermediate Algebra, 3edSection 1.4
Find the slope of the line
Slide 4
Finding Slope from a Linear Equation
3 8.y x
x y
0 8
1 5
2 2
3 –1
Create a table using x = 1, 2, 3. Then sketch the graph.
rise 33
run 1m
Example
Solution
Finding Slope from a Linear Equation
Lehmann, Intermediate Algebra, 3edSection 1.4
For a linear equation of the form , m is the slope of the line.
Slide 5
Finding Slope from a Linear Equation
y mx b
Are the lines parallel,
perpendicular, or neither?
53 and 12 10 5
6y x y x
Property
Example
Finding Slope from a Linear Equation
Lehmann, Intermediate Algebra, 3edSection 1.4
• For the line the slope is
• For the other equation we solve for y:
Slide 6
Finding Slope from a Linear Equation
53
6y x 5
6
12 10 5
12 10 10 5 10
12 10 5
12 10 512 12 12
5 56 12
y x
y x x x
y x
yx
y x
Original EquationAdd 10x to both sides.Combine & rearrange terms
Divide both sides by 12.
Simplify.
Property
Finding Slope from a Linear Equation
Lehmann, Intermediate Algebra, 3edSection 1.4
• For the line the slope is
• Since the slopes are the same for both equations, the lines are parallel
Slide 7
Finding Slope from a Linear Equation
5 56 12
y x 5
12
We use ZStandard followed by ZSquare to draw the line in the same coordinate system.
Solution Continued
Graphing Calculator
Finding Slope from a Linear Equation
Lehmann, Intermediate Algebra, 3edSection 1.4
For the line , if the run is 1, then the rise is m.
Slide 8
Vertical Change Property
y mx b
Vertical Change property for a negative slope.
Vertical Change property for a positive slope.
Property
Vertical Change Property
Lehmann, Intermediate Algebra, 3edSection 1.4
Sketching Equations: • It’s helpful to know the y-intercept.• y-intercept has a x-value of 0.• Substitute x = 0 gives
Slide 9
Finding the y-intercept of a Linear Line
0y m b b
For a linear equation of the form , the y-intercept is (0, b).
y mx b Property
Finding the y-Intercept of linear Equation
Lehmann, Intermediate Algebra, 3edSection 1.4
What is the y-intercept of
Slide 10
Finding the y-intercept of a Linear Line
53?
6y x
• b is equal to 3, so the y-intercept is (0, 3)
If an equation of the form , we say that it is in slope-intercept form.
y mx b
Example
Solution
Definition
Finding the y-Intercept of linear Equation
Lehmann, Intermediate Algebra, 3edSection 1.4 Slide 11
Graphing Linear Equations
Sketch the graph of y = 3x – 1.
• The y-intercept is (0, –1) and the slope is 3 rise
31 run
To graph:
1.Plot the y-intercept, (0, 1). (continued)
Example
Solution
Graphing Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.4 Slide 12
Graphing Linear Equations
2. From (0, –1), look 1 unit to the right and 3 units up to plot a second point, which we see by inspection is (1, 2).
3. Sketch the line that contains these two points.
Solution Continued
Graphing Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.4 Slide 13
Graphing Linear Equations
To sketch the graph of a linear equation of the form
1.Plot the y-intercept (0, b).
2.Use m = to plot a second point.
3.Sketch the line that passes through the two plotted points.
riserun
y mx b
Guidelines
Graphing Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.4 Slide 14
Graphing Linear Equations
Sketch the graph of 2x + 3y = 6.
First we rewrite into slope-intercept form:2 3 6
2 3 2 6 2
3 2 6
3 2 63 3 3
x y
x y x x
y x
yx
Original EquationSubtract 2x from both sides.Combine & rearrange terms
Divide both sides by 3.
Example
Solution
Graphing Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.4 Slide 15
Graphing Linear Equations
y-intercept: (0, 2) Slope:
1. Plot the y-intercept, (0, 2).
2. From the point (0, 2), look 3 units to the right and 2 units down to plot a second point, which we see by inspection is (3, 0).
2 2 rise3 3 run
22
3y x -a a
Simplfy : =-b b
Solution Continued
Graphing Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.4 Slide 16
Graphing Linear Equations
3. Then sketch the line that contains these two points. We can verify our result by checking that both (0, 2) and (3, 0) are solutions.
Solution Continued
Graphing Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.4 Slide 17
Graphing Linear Equations
1. Determine the slope and the y-intercept 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.
First we rewrite into slope-intercept form:
Example
Solution
Graphing Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.4 Slide 18
Graphing Linear Equations
Slope is and the y-intercept is
ax by c
ax by ax c ax
by ax c
by a cx
b b ba c
y xb b
Divide both sides by b.
-a aSimplfy : =-
b b
Original equation
Subtract ax from both sides.
Combine and rearrange terms.
0, .cb
ab
Solution Continued
Graphing Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.4 Slide 19
Graphing Linear Equations
Given that ax + by = c in slope-intercept form is .
3 c 5 and the y-intercept, 0, 0, .
7 b 7ba
a cy x
b b , then given 3x + 7y = 5, we substitute .
3 for a, 7 for b and 5 for c. Thus, the slope, .
Solution Continued
Graphing Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.4
For the following sets, is there a line that passes through them? If so, find the slope of that line.
Slide 20
Slope Addition Property
• Value of x increases by 1.
• Value of y changes by –3.
•The slope is –3.
Example
Solution
Slope Addition Property
Lehmann, Intermediate Algebra, 3edSection 1.4 Slide 21
Slope Addition Property
Set 2 • Value of x increases by 1. • Value of y changes by 5.
So, the slope is 5.
Set 3• Value of x increases by 1. •Value of y does not change by the same value. Hence, not a line.
Solution Continued
Slope Addition Property