4-6 Writing Linear Equations
for Word Problems
Give a problem that has y-intercept like in the
sample problem. Problems in alignment with
handout.
Algebra 1
Glencoe McGraw-Hill
Linda Stamper
There are two basic types of real-life problems that
can be solved with linear equations.
Type I Problems involving a constant rate of change (slope).
(Write equation in slope-intercept form.)
There are two basic types of real-life problems that
can be solved with linear equations.
Type I Problems involving a constant rate of change (slope).
(Write equation in slope-intercept form.)
Type II Problems involving two variables, x and y, such that the
sum of Ax + By is a constant.
(Write equation in standard form.)
You open a savings account with $150. You plan to add $50 each
month. Write an equation that gives the monthly savings, y (in dollars),
in terms of the month, x.
savings (dollars)
y
No graph is given.
•
Can you visualize the
graph in your head?
(5,400)
•
(3,300)
If the y-intercept is
not given, what piece
of information would
you need? a point
•
100
50
0 1 2
What is the
equation of this
y  50
x  150
line?
months
x
If the slope is not
given, what
information would you
need?
two points
Between 1980 and 1990, the monthly rent for a one-bedroom
apartment increased by $40 per year. In 1985, the rent was $300
per month. Write an equation that gives the monthly rent y (in
dollars), in terms of the year x. Let year 0 correspond to 1980.
Between 1980 and 1990, the monthly rent for a one-bedroom
apartment increased by $40 per year. In 1985, the rent was $300
per month. Write an equation that gives the monthly rent y (in
dollars), in terms of the year x. Let year 0 correspond to 1980.
Assign Labels:
Let x = years
Let y = monthly rent in dollars
Find the constant rate of change (slope). m  40
Think of a point on the graph that represents 1985 and $300.
y
Write an equation.
y  mx  b
300  405  b
300  200  b
y  300  40x  200
 200  200
 300
 300
100  b
y  40x  100
y  40x  100
y  300  40 x  5 
rent (dollars)
y  y1  mx  x1 
•
(5,300)
x1 , y1

1980
years
x
Recall value problems.
Andrea has 52 coins in dimes and quarters worth $10. How many of
each coin does she have?
Number Labels and Value Labels
Let d = number of dimes
Let 52 – d = number quarters
Let .10d = value of the dimes
Let .25(52 – d) = value of quarters
Verbal Model.
Write the equation.
Value of dimes + Value of quarters = Total Value
.10d + .25(52 – d) = 10
Today you will write a linear equation that has two variables to
represent a word problem.
Type II Problems involving two variables, x and y, such that the sum
of Ax + By is a constant.
Mrs. Burke’s drama class is presenting a play next Friday night and you
have been put in charge of ticket sales. Tickets for adults are sold for
$4 each and students pay $2 each. Write a linear equation, in standard
form, that models the different number of adult tickets, x, and the
number of student tickets, y, that could be sold if total ticket sales are
$320.
Mrs. Burke’s drama class is presenting a play next Friday night and you
have been put in charge of ticket sales. Tickets for adults are sold for
$4 each and students pay $2 each. Write a linear equation, in standard
form, that models the different number of adult tickets, x, and the
number of student tickets, y, that could be sold if total ticket sales are
$320.
Number Labels and Value Labels
Let x = number of adult tickets
Let y = number of student tickets
Let 4x = value of adult tickets
Let 2y = value of student tickets
Verbal model.
Value of adult + value of student = total value of
tickets
tickets
tickets
Write the equation.
4x  2y  320
2x  y  160
Ax  By  C
Example 1 Between 1980 and 1990, the monthly rent for a onebedroom apartment increased by $20 per year. In 1987, the rent was
$350 per month. Write an equation that gives the monthly rent y (in
dollars), in terms of the year x. Let year 0 correspond to 1980.
Example 1 Between 1980 and 1990, the monthly rent for a onebedroom apartment increased by $20 per year. In 1987, the rent was
$350 per month. Write an equation that gives the monthly rent y (in
dollars), in terms of the year x. Let year 0 correspond to 1980.
Assign Labels
Let x = years
Let y = monthly rent in dollars
m  20
Find the constant rate of change (slope).
Think of a point on the graph that represents 1987 and $350.
y
Write an equation.
y  mx  b
350  207   b
350  140  b
y  350  20x  140
 140
 350  140
 350
210  b
y  20x  210
y  20x  210
•
(7,350)
x1 , y1

rent (dollars)
y  y1  mx  x1 
y  350  20 x  7 
1980
years
x
Example 2 You are flying from LA to Boston. Three hours into the
trip you have traveled 825 miles. You are traveling at an average
speed of 275 mph. Write a linear equation that gives the distance y
(in miles), after x hours of flying.
Example 2 You are flying from LA to Boston. Three hours into the
trip you have traveled 825 miles. You are traveling at an average
speed of 275 mph. Write a linear equation that gives the distance y
(in miles), after x hours of flying.
Assign Labels
Let x = hours flying
Find the constant rate of change (slope).
Let y = distance in miles
m  275
Think of the point on a graph that represents 3hrs and 825 mi.
y
Use point-slope to write an equation.
y  y1  mx  x1 
y  825  275x  3
y  mx  b
825  2753  b
y  825  275 x  825 825  825  b
 825  825  825
 825
0b
y  275 x
y  275 x
d
i
s
t
a
n
c
e
•
(3,825)
x1 , y1
hours

x
Example 3 By the end of your 5th French lesson you have learned 20
vocabulary words. After 10 lessons you know 40 vocabulary words.
Write an equation that gives the number of vocabulary words you know y,
in terms of the number of lessons you have had x.
Assign Labels
Find the slope.
5,20  10,40 
m


y2  y1
x2  x1
20  40
5  10
 20
5
4
Let x = # lessons
Let y = # vocab. words
Write an equation.
y  y1  mx  x1 
y  40  4x  10 
y  40  4x  40
 40
 40
y  4x
Example 4 Joan earns $4 an hour when she works on Fridays, and $6
an hour on Saturdays. Write a linear equation that models the
different number of hours she could have worked on each of the two
days if she earned $36. If x (Friday’s hours) and y (Saturday’s hours)
are whole number, find all the possible combinations she could have
worked on Friday and Saturday.
Example 4 Joan earns $4 an hour when she works on Fridays, and $6
an hour on Saturdays. Write a linear equation that models the
different number of hours she could have worked on each of the two
days if she earned $36. If x (Friday’s hours) and y (Saturday’s hours)
are whole number, find all the possible combinations she could have
worked on Friday and Saturday.
Assign number labels.
Assign earning labels.
Verbal model.
Let x = Fri. hours
Let 4x = Fri. earn.
Let y = Sat. hours
Let 6y = Sat. earn.
Friday’s earn. + Saturday’s earn. = total earn.
Write the equation.
4x  6y  36
2x  3y  18
Ax  By  C
Example 4 Joan earns $4 an hour when she works on Fridays, and $6
an hour on Saturdays. Write a linear equation that models the
different number of hours she could have worked on each of the two
days if she earned $36. If x (Friday’s hours) and y (Saturday’s hours)
are whole number, find all the possible combinations she could have
worked on Friday and Saturday.
Make a table.
y
2x  3y  18
x
y
0
9
6
3
6
0
2
4
S
a
t
.
h
o
u
r
s
(0,6)
•
•
(3,4)
(6,2)
•
Friday hours
(9,0)
•
x
Example 5 You want to buy some CDs and Videos for the Teen Center
recreation room. You have $75 to spend. You can order CDs for $15
each and Videos for $5 each from your discount catalog. Write a linear
equation that models the different number of CDs and Videos you could
purchase. If x (number of CDs) and y (number of Videos) are whole
numbers, find all the possible combinations you could purchase.
Assign # labels.
Let x = # CDs
Let y = # Videos
Assign cost labels.
Let 15x = cost of CDs
Let 5y = cost of Videos
Verbal model.
Cost of CDs + Cost of Videos = total cost
Write the equation.
15x  5 y  75
3x  y  15
Ax  By  C
y
Make a table and graph.
3x  y  15
y
0
15
5
0
1
12
2
9
3
6
4
3
•
# of Videos
x
•
•
•
•
•
# of CDs
The ordered pairs represent the possible combinations.
(0,15), (5,0), (1,12), (2,9), (3,6), (4,3)
x
Practice 1 You collect comic books. In 1990 you had 60 comic books. By
the year 2000 you had 240 comic books. Find an equation that gives the
number of comic books y, in terms of the year, x. Let year 0 correspond
to 1990.
Practice 2 Maria decides to start jogging every day at the track.
The first week she jogs 4 laps. She plans to increase her laps by one
lap per week. Let y represent the number of laps Maria runs and let x
represent the number of weeks she’s running. Write a linear equation
that gives the number of laps she is running after x weeks.
Practice 1 You collect comic books. In 1990 you had 60 comic books. By
the year 2000 you had 240 comic books. Find an equation that gives the
number of comic books y, in terms of the year, x. Let year 0 correspond
to 1990.
Assign Labels
Let x = year
Identify two points.
0,60  and 10,240 
y  y1
m 2
x2  x1


60  240
0  10
 180
 10
 18
Let y = # comic books
Use slope-intercept to write an equation.
y  mx  b
y  18x  60
Practice 2 Maria decides to start jogging every day at the track.
The first week she jogs 4 laps. She plans to increase her laps by one
lap per week. Let y represent the number of laps Maria runs and let x
represent the number of weeks she’s running. Write a linear equation
that gives the number of laps she is running after x weeks.
Assign Labels
Find the slope.
Let x = # weeks
y  y1  mx  x1 
1
Identify a point.
Use point-slope to
write an equation.
Let y = # laps
1,4 
y  4  1 x  1
y  4  x 1
4
4
y  x3
4-A13 Handout A13