Download B-4 Writing Linear Equations

Linear Relations and
Functions
B-4
Writing Linear Equations
ACT WARM-UP
Ten thousand tires are being stored in a
warehouse. Two percent of the tires are
not usable. What is the ratio of usable
tires to not usable tires?
A)
1
98
B)
1
50
C)
1
49
D)
49
1
E)
50
1
If 2% are not usable, then 98% are usable.
49
Therefore 98:2 is simplified to D) .
1
Objectives
Write an equation of a line given the slope
and a point on the line
Write an equation of a line parallel or
perpendicular to a given line
Essential Question
How do you write an equation of a
line given the coordinates of two
points on a line?
Slope-Intercept Form of a Linear
Equation
The slope-intercept form of the equation of
a line is y = mx + b, where m is the slope
and b is the y-intercept.
y = mx + b
slope
y-intercept
If you are given the slope and y-intercept of a
line, you can quickly find an equation of the
line by substituting the values of m and b into
the slope-intercept form. For example, if you
know that the slope of a line is – 3 and the
y-intercept is 4, the equation of the line is
y = – 3x + 4, or, in standard form, 3x + y = 4.
In previous courses you have also learned
that the slope-intercept form can be used to
find an equation of a line given the slope and
the coordinates of any point on the line.
Write an equation in slope-intercept form for the line
that has a slope of
and passes through (5, –2).
Slope-intercept form
–
Simplify.
Add 3 to each side.
Answer: The y-intercept is 1. So, the equation in
slope-intercept form is
More often in algebra, when you are
given slope and a point or two points
on a line most students find it difficult
and tedious to solve for the y-intercept
(b) in the slope-intercept form. An
easier method to use given slope and
a point or given two coordinates on a
line is the point-slope form.
Point-Slope Form of a Linear
Equation
The point-slope form of the equation of a
line is y – y1 = m(x – x1), where (x1,y1) are
the coordinates of a point on the line and
m is the slope of the line.
y – y1 = m(x – x1)
slope
coordinates of point on line
Write an equation of the line that passes
through (5, 4) and has a slope of −3.
Solution
Because you know the slope and a point on the
line, use point-slope form to write an equation of
the line Let 𝑥1 , 𝑦1 = 5, 4 and m = −3.
𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1
𝑦 − 4 = −3 𝑥 − 5
𝑦 − 4 = −3𝑥 + 15
𝑦 = −3𝑥 + 19
Use point-slope form
Substitute for m, 𝑥1 , 𝑎𝑛𝑑 𝑦1
Distributive property
Write in slope-intercept form.
Write an equation in slope-intercept form for the line
that has a slope of
Answer:
and passes through (–3, –1).
Multiple-Choice Test Item
What is an equation of the line through (2, –3)
and (–3, 7)?
A
B
C
D
Read the Test Item
You are given the coordinates of two points on the line.
Notice that the answer choices are in slope-intercept form.
Solve the Test Item
First, find the slope of the line.
Slope formula
Simplify.
The slope is –2. That eliminates choices
B)
and C)
.
Then use the point-slope formula to find an equation.
Point-slope form
you can use either
point for
.
Distributive Property
Subtract 3 from each side.
Answer: D
Multiple-Choice Test Item
What is an equation of the line through (2, 5)
and (–1, 3)?
A
C
B
D
Answer: C
When changes in real-world situations occur at a linear rate, a linear equation
can be used as a model for describing the situation.
Sales As a part-time salesperson, Jean Stock is paid
a daily salary plus commission. When her sales are
$100, she makes $58. When her sales are $300, she
makes $78.
Write a linear equation to
model this situation.
Let x be her sales and let y be
the amount of money she makes.
Use the points (100, 58) and
(300, 78) to make a graph to
represent the situation.
Slope formula
Simplify.
Now use the slope and either of the given points with the
point-slope form to write the equation.
Point-slope form
Distributive Property
Add 58 to each side.
Answer: The slope-intercept form of the equation
is
What are Ms. Stock’s daily
salary and commission rate?
The y-intercept of the line is 48.
The y-intercept represents the
money Jean would make if she
had no sales. Thus, $48 is her
daily salary.
The slope of the line is 0.1. Since
the slope is the coefficient of x,
which is her sales, she makes
10% commission.
Answer: Ms. Stock’s daily salary is $48, and she makes
a 10% commission.
How much would Jean make in a day if her sales
were $500?
Find the value of y when
Use the equation you found
in Example 3a.
Replace x with 500.
Simplify.
Answer: She would make $98 if her sales were $500.
Sales The student council is selling coupon books to
raise money for the Humane Society. If the group
sells 200 books, they will receive $150 dollars. If they
sell 500 books, they will make $375.
a. Write a linear equation to model this situation.
Answer:
b. Find the percentage of the proceeds that the student
council receives.
Answer: 75%
c. If they sold 1000 books, how much money would
they receive to donate to the Humane Society?
Answer: $750
The slope-intercept and point-slope forms can be used to find equations of lines
that are parallel or perpendicular.
Write an equation for the line that passes through
(3, –2) and is perpendicular to the line whose
equation is
The slope of the given line is –5. Since the slopes of
perpendicular lines are opposite reciprocals, the slope
of the perpendicular line is
Use the point-slope form and the ordered pair (3, –2)
to write the equation.
Point-slope form
Distributive Property
Add –2 to each side.
Answer: An equation of the line is
Write an equation for the line that passes through
(3, 5) and is perpendicular to the line whose
equation is
Answer:
Essential Question
How do you write an equation of a
line given the coordinates of two
points on a line?
Find the slope using the slope formula and use the pointslope form, y – y1 = m(x – x1), to substitute the slope for m
and one of the given points for (x1,y1).
Math Humor
Student: I’ll just draw a quick line by hand
and guess the slope.
Teacher: No, no. It’s point-slope form, not
point-sloppy form.