Writing and Graphing Equations of Lines
Use the slope-intercept form of the equation of a line.
Graph a line by using its slope and a point on the line.
Write an equation of a line by using its slope and any point on the line.
Write an equation of a line by using two points on the line.
Write an equation of a line that fits a data set.
3.4
2
3
4
5
1
Objective 1
Use the slope-intercept form of the equation of a line.
Slide 3.4-3
Use the slope-intercept form of the equation of line.
In Section 3.3, we found the slope of a line by solving for y. In that form, the slope is the coefficient of x. For, example, the slope of the line with equation y = 2x + 3 is 2. So, what does 3 represent?
Suppose a line has a slope m and y-intercept (0,b). We can find an equation of this line by choosing another point (x,y) on the line as shown. Then we use the slope formula.
Slide 3.4-4
mx b y
mx y b
y bm
x
y mx b
0
y bm
x
Rewrite.
Add b to both sides.
Multiply by x.
Subtract in the denominator.
Change in x-values
Change in y-values
Use the slope-intercept form of the equation of line.
The result is the slope-intercept form of the equation of a line, because both the slope and the y-intercept of the line can be read directly from the equation. For the line with the equation y = 2x + 3, the number 3 gives the y-intercept (0,3).
Slope-Intercept FormThe slope-intercept form of the equation of a line with slope m and y-intercept (0,b) is
Where m is the slope and b is the y-intercept (0,b).
Slide 3.4-5
y m bx
Identify the slope and y-intercept of the line with each equation.
Solution:
y x
Slide 3.4-6
EXAMPLE 1 Identifying Slopes and y-Intercepts
36
4y x
Slope: − 1
y-intercept: (0,0)
Slope:
y-intercept: (0,− 6)
3
4
Write an equation of the line with slope −1 and y-intercept (0,5).
Solution:
5y x
Slide 3.4-7
EXAMPLE 2 Writing an Equation of a Line
Objective 2
Graph a line by using its slope and a point on the line.
Slide 3.4-8
Graph a line by using its slope and a point on the line.
Graphing a Line by Using the Slope and y-Intercept
Step 1: Write the equation in slope-intercept form, if necessary, by solving for y.
Step 2: Identify the y-intercept. Graph the point (0,b).
Step 3: Identify slope m of the line. Use the geometric interpretation of slope (“rise over run”) to find another point on the graph by counting from the y-intercept.
Step 4: Join the two points with a line to obtain the graph.
Slide 3.4-9
Solution:
33 34 8xyx x
Graph 3x – 4y = 8 by using the slope and y-intercept.
4 3
4 4 4
8xy
32
4y x
Slope intercept form
Slide 3.4-10
EXAMPLE 3 Graphing Lines by Using Slopes and y-Intercepts
Solution:
Graph the line through (2,−3) with slope1.3
Make sure when you begin counting for a second point you begin at the given point, not at the origin.
Slide 3.4-11
EXAMPLE 4 Graphing a Line by Using the Slope and a Point
Objective 3
Write an equation of a line by using its slope and any point on the line.
Slide 3.4-12
Write an equation of a line by using its slope and any point on the line.
We can use the slope-intercept form to write the equation of a line if we know the slope and any point on the line.
Slide 3.4-13
Solution:
y m bx
Write an equation, in slope-intercept form, of the line having slope −2 and passing through the point (−1,4).
4 2 1 b
4 22 2b
2b
2 2y x The slope-intercept form is
Slide 3.4-14
EXAMPLE 5 Using the Slope-Intercept Form to Write an Equation
Write an equation of a line by using its slope and any point on the line.
There is another form that can be used to write the equation of a line. To develop this form, let m represent the slope of a line and let (x1,y1) represent a given point on the line. Let (x, y) represent any other point on the line.
Point-Slope FormThe point-slope form of the equation of a line with slope m passing through point (x1,y1) is
11 1
1
m xy
xy
xx
xx
11 yx xm y
1 1 .m xy xy
Slope
Given point
1
1
my y
x x
Slide 3.4-15
Multiply each side by x − x1.
Definition of slope
Rewrite.
Solution:
Write an equation of the line through (5,2), with the slope Give the final answer in slope-intercept form.
1.3
1 1my xy x
12 5
3y x
1 52
3
6
32
3y x
1 11
3 3xy
Slide 3.4-16
EXAMPLE 6 Using the Point-Slope Form to Write Equations
Objective 4
Write an equation of a line by using two points on the line.
Slide 3.4-17
Many of the linear equations in Section 3.1−3.3 were given in the form
called standard form, where A, B, and C are real numbers and A and
B are not both 0.
,Ax B Cy
Slide 3.4-18
Write an equation of a line by using two points on the line.
Find an equation of the line through the points (2,5) and (−1,6). Give the final answer in slope-intercept form and standard form.
Solution:
2 1
2 1x
ym
y
x
The same result would also be found by substituting the slope and either given point in slope-intercept form and then solving for b.
6 5
1 2m
1
3m
1 1my xy x
1
36 1xy
1 17
3 3xy
1 16
3
18
36
3xy
Slide 3.4-19
EXAMPLE 7 Writing the Equation of a Line by Using Two Points
173x y
3 17y x
1 17
3 3xy
Standard form
Slope-intercept form
Slide 3.4-20
Summary of the forms of linear equations.
Objective 5
Write an equation of a line that fits a data set.
Slide 3.4-21
Solution:
Use the points (3, 4645) and (7, 6185) to write an equation in slope-intercept form that approximates the data of the table. How well does this equation approximate the cost in 2005?
2 1
2 1x
ym
y
x
6185 4645
7 3m
0
4
154m
1 1my xy x 4645 8 33 5 xy
385 3490xy
4645 385 11554645 4645xy
385m
The equation gives y = 5415 when x = 5, which is a very good approximation.
Slide 3.4-22
EXAMPLE 8 Writing an Equation of a Line That Describes Data
(5)385 3490y 5415y