Summer 2016 - Session 2
Math 1300
FUNDAMENTALS OF MATH
Section #16535
Monday - Friday, 10am-12pm
Instructor: Dr. Angelynn Alvarez
[email protected]
Section 2.4 – Equations of Lines
Section 2.4 – Equations of Lines
There are several ways to represent a line through an equation.
In this class (and in college algebra), we will discuss 3 different
forms of an equation of a line:
(1)
The standard form of a line is an equation of the form
𝑎𝑥 + 𝑏𝑦 = 𝑐
where 𝑎, 𝑏, 𝑐 are numbers.
(2)
The point-slope form of a line is an equation of the form
𝑦 − 𝑦! = 𝑚 𝑥 − 𝑥!
where (𝑥! , 𝑦! ) is a point on the line and 𝑚 is the slope.
(3)
The slope-intercept form of a line is an equation of the
form
𝑦 = 𝑚𝑥 + 𝑏
where m is the slope and b is the 𝑦-intercept.
In summary:
• standard form – has both variables 𝑥 and 𝑦 on the left of “=”
• point slope form – has parentheses
• slope-intercept form – has 𝑦 by itself on the left of “=”
Our “favorite” and most useful form will be slope-intercept form since
one can easily read off what the slope is and what the y-intercept is.
Because slope-intercept form is deemed a favorite, you will be asked
to rewrite equations into slope-intercept form.
Writing Equations in Slope-Intercept Form:
Ø GOAL: Solve for 𝑦 (get 𝑦 by itself on the left side of “=”).
*Important (for this section):
If 𝑎 and 𝑏 are numbers and 𝑥 is the variable, then
𝒂𝒙 𝒂
= 𝒙
𝒃
𝒃
Example: Give the slope-intercept form of the following lines.
−2𝑥 − 4𝑦 = 5
𝑥 + 2𝑦 = 7
3𝑥 + 5𝑦 = −1
−𝑥 + 6𝑦 = −3
Recall: If an equation is in slope-intercept form, one can easily read
off what the slope is and what the 𝑦-intercept is.
Determining the slope and y-intercept (given an equation):
(1)
Rewrite the equation in slope-intercept form (solve for 𝑦).
* If the equation is already in slope-intercept form, skip
this step.
(2)
The number being multiplied by 𝑥 is the slope.
(3)
The number after the 𝑥 is the y-intercept.
*Subtraction → the 𝑦-intercept is negative.
Example: Find the slope and the 𝑦-intercept of the following lines.
2𝑥 − 3𝑦 = −2
5𝑥 + 4𝑦 = −2
−6𝑥 − 3𝑦 = −3
Example: Find the slope (𝑚) and the 𝑦-intercept (𝑏) of the following
lines.
4𝑥 − 2𝑦 = 6
6𝑥 − 4𝑦 = −5
−3𝑥 − 2𝑦 = 7
Writing an equation of a line in slope-intercept form given its
slope and its 𝒚-intercept:
Ø
Substitute the slope (𝑚) and the y-intercept (𝑏) into the
equation 𝒚 = 𝒎𝒙 + 𝒃. Then simplify if needed.
Example: Give an equation in slope-intercept form for the line with
slope
!
!
and 𝑦-intercept 3.
Example: Give an equation in slope-intercept form for the line with
!
slope − and 𝑦-intercept −5.
!
Example: Give an equation in slope-intercept form for the line with
slope
!
!
and 𝑦-intercept −4.
Writing an equation of a line in slope-intercept form given its
slope and a general point on the line (𝒙, 𝒚):
Goal: We want to write an equation 𝒚 = 𝒎𝒙 + 𝒃, where 𝑚 and 𝑏 are
numbers and 𝑥 and 𝑦 are variables.
(1)
(2)
(3)
Substitute 𝑚, 𝑥, and 𝑦 into the equation 𝑦 = 𝑚𝑥 + 𝑏.
Solve for 𝑏.
Substitute 𝑚 and 𝑏 into 𝑦 = 𝑚𝑥 + 𝑏 (and leave 𝑥 and 𝑦 as
variables).
Example: Give an equation in slope-intercept form for the line with
slope 3 and passes through the point (−5, −4).
Example: Give an equation in slope-intercept form for the line with
slope −5 and passes through the point (2, −3).
Example: Give an equation in slope-intercept form for the line with
slope −1 and passes through the point (−4, 4).
Example: Give an equation in slope-intercept form for the line with
slope −2 and passes through the point (5, 6).
Writing an equation of a line in slope-intercept form given 2
points on the line: 𝒙𝟏 , 𝒚𝟏 and 𝒙𝟐 , 𝒚𝟐 :
(1)
Find the slope 𝑚: Compute the slope by using the slope
formula:
𝑦! − 𝑦!
𝑚=
𝑥! − 𝑥!
(2)
(3)
Find the 𝑦-intercept 𝑏: Pick 1 of the given points and
substitute slope (𝑚) and the values for 𝑥 and 𝑦 into the
equation 𝑦 = 𝑚𝑥 + 𝑏. Then solve for 𝑏.
Substitute 𝑚 and 𝑏 into 𝑦 = 𝑚𝑥 + 𝑏 (and leave 𝑥 and 𝑦 as
variables).
Example: Write an equation in slope-intercept form for the line that
passes through the points (6, 0) and (5, −4).
Example: Write an equation in slope-intercept form for the line that
passes through the points (−3, 0) and (2, −7).
Example: Give the equation (in slope-intercept form) of the line that
passes through the points −2, 3 and (4, 1).
Example: Give the equation (in slope-intercept form) of the line that
passes through the points 2, 2 and (1, −3).
Example: Give the equation (in slope-intercept form) of the line that
passes through the points −4, −3 and (2, −1).
Example: Give the equation (in slope-intercept form) of the line that
passes through the points 5, −1 and (−3, 4).
Example: The 𝑥-intercept of the line is (−2, 0). The 𝑦-intercept of
the line is (0, 5). Find an equation of the line in slope-intercept form.
Example: The 𝑥-intercept of the line is (3, 0). The 𝑦-intercept of the
line is (0, −1). Find an equation of the line in slope-intercept form.