3.7 Equations of Lines in theCoordinate Plane
SOL G3a
Objectives: TSW …
• investigating and calculating slopes of a line given two points
on the line.
• write the equation of a line in slope-intercept form.
• write the equation of a line in point-slope form.
Slope (m)
The ratio of its vertical rise to its horizontalrun.
Steepness
Slope = m =
Vertical rise
Horizontal run
Example 1:
Find the slopes.
-8
2
2
4
Slope (continued)
of a line containing two points withcoordinates (x1, y1) and (x2, y2) isgiven by the formula
Slopes
All horizontallines have a 0slope
All vertical lines
have anundefinedslope
Positive Slopes
Rise (upward) as you move left to right
Line slopesup from leftto right
y
x
Negative Slope
Fall (downward) as you move left to right
Line slopesdown fromleft to right
y
x
Example 2:
Find the slope using theslope formula.
Rate of Change
Describes how a quantity is changingover time.
The slope of a line can be used todetermine the Rate of Change
Change in quantity (y)
Change in time (x)
Example 3:
Recreation: For one manufacturer of camping equipment,
between 1990 and 2000 annual sales increased by $7.4 million
per year. In 2000, the total sales were $85.9 million. If the
sales increase at the same rate, what will be the total sales in
2010?
+85.9 +85.9
159.9 mill. = y2
7
4
.
0
=
y
2
–
8
5
.
9
7.4(10) = y2 – 85.9
Forms of Linear Equations
Slope-Intercept Form -
y = mx + b
slope
y-intercept
Point-Slope Form -
y – y1 = m(x – x1)
slope
x-coordinate
y-coordinate
Example 4: Graph
1.) The equation is in slope-intercept form y = mx + b
The slope is
y-intercept (0, 1)
2.) Plot the point (0, 1)
3.) Use the slope , from
the point (0, 1) go up 2,
right 3
Example 5: Graph
1.) The equation is in slope-intercept form y = mx + b
The slope is 3
y-intercept (0, 1)
2.) Plot the point (0, 1)
3.) Use the slope 3, from
the point (0, 1) go up 3,
right 1
Example 6: Graph y - 3 = -2(x + 3)
1.) The equation is in point-slope form y – y1 = m(x – x1)
The slope is -2
Point on line (-3, 3)
2.) Plot the point (-3, 3)
3.) Use the slope -2, from
the point (-3, 3) go
down 2, right 1
Example 7: Graph
Point on line (4, 2)
2.) Plot the point (4, 2)
The slope is
3.) Use the slope , from
the point (4, 2) go
down 1, right 3
1.) The equation is in point-slope form y – y1 = m(x – x1)
Writing Equations of Linear Lines
If we know the slope and at least onepoint
If we have the slope and y-intercept, usethe slope-intercept form; y = mx + b
If we have the slope and a point, use thepoint-slope form; y – y1 = m(x – x1)
Example 8: Write the equation of the line
What is an equation of the line with slope3 and y-intercept -5?
Start with the slope-intercept form ofthe equation
y = mx + b
y = 3x + (-5)
Substitute 3 for m, and -5for b
Simplify
y = 3x - 5
Example 9: Write the equation of the line
What is an equation of the line throughpoint (-1, 5) with slope 2?
Start with the point-slope form of theequation
y – y1 = m(x – x1)
y – 5 = 2(x - (-1))
Substitute 2 for m, and -1in for x1 and 5 in for y1
Simplify
y – 5 = 2(x + 1)
Example 10: Write the equation of the line
What is an equation of the line with
slope and y-intercept 2?
Start with the slope-intercept form ofthe equation
y = mx + b
Substitute for m, and2 for b
y = x + 2
Example 11: Write the equation of the line
What is an equation of the line throughpoint (-1, 4) with slope -3?
Start with the point-slope form of theequation
y – y1 = m(x – x1)
y – 4 = -3(x - (-1))
Substitute -3 for m, and -1in for x1 and 4 in for y1
Simplify
y – 4 = -3(x + 1)
Writing Equations of Linear Lines
If we know two points on the line
Find the slope using the formula
Using the point-slope formula
Plug in one of the two points
Plug in the slope for m
Example 12:Write the equation of the line
What is an equation of the line through point (-2, -1)and point (3, 5)?
Find the slope
y + 1 = (x + 2) or
y - 3 = (x - 5)
Start with the point-slope form of the equation
y – y1 = m(x – x1)
Plug in the slope and one of the two points
Writing Equations Horizontal andVertical Lines
We don’t need a slope
All points on a horizontal line have thesame y-coordinate; so the equation isy = y1.
All points on a vertical line have thesame x-coordinate; so the equation isx = x1.
Where (x1, y1)
Example 13:Write the equation of the line
What are the equations for the horizontal andvertical lines through (2, 4)?
The horizontal is y = y1
y = 4
Substitute 4 for y1
The vertical is x = x1
x = 2
Substitute 2 for x1
Example 14: Write the equation ofthe line
What are the equations for the horizontal andvertical lines through (4, -3)?
The horizontal is y = y1
y = -3
Substitute -3 for y1
The vertical is x = x1
x = 4
Substitute 4 for x1
Homework
Pg 194 – 195 # 9 – 37 odds