4-6 Writing Linear Equations for Word

Problems

Algebra 1 Glencoe McGraw-Hill Linda Stamper

Give a problem that has y-intercept like in the sample problem. Problems in alignment withhandout.

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.)

x

y

•

months

savin

gs

(dollars

) (3,300)

50100

1 2

••(5,400)

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.

No graph is given.Can you visualize the graph in your head?If the y-intercept is not given, what piece of information would you need?

What is the equation of this

line?150x5y 0

If the slope is not given, what information would you need?

a point

two points

0

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.

Find the constant rate of change (slope).

x

y

•

40mAssign Labels: Let x = years Let y = monthly rent in dollars

Think of a point on the graph that represents 1985 and $300.

years

ren

t (d

ollars

)1980

(5,300)

Write an equation.

11 xxmyy 5x40300y

20040x300y 300 300

10040xy

11

y,x

bmxy b540300

b200300 002 200

b100 10040xy

Let d = number of dimes

Andrea has 52 coins in dimes and quarters worth $10. How many of each coin does she have?

Let .10d = value of the dimes

.10d + .25(52 – d) = 10

Let 52 – d = number quarters

Number Labels and Value Labels

Let .25(52 – d) = value of quarters

Verbal Model. Value of dimes + Value of quarters = Total Value

Write the equation.

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.

Recall value problems.

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.

Verbal model. Value of adult + value of student = total value of tickets tickets ticketsWrite the equation. 320y2x4

CyBxA

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.

Let x = number of adult ticketsLet 4x = value of adult tickets

Let y = number of student tickets

Number Labels and Value Labels

Let 2y = value of student tickets

160yx2

Example 1 Between 1980 and 1990, the monthly rent for a one-bedroom 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 one-bedroom 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.

Find the constant rate of change (slope).

x

y

•

20mAssign Labels Let x = years Let y = monthly rent in dollars

Think of a point on the graph that represents 1987 and $350.

years1980

(7,350)

Write an equation.

11 xxmyy 7x20350y

14020x350y 350 350

21020xy

11

y,x

ren

t (d

ollars

)

bmxy b720350

b140350 140 401

b210 21020xy

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.

Find the constant rate of change (slope).

x

y

•

275m

Assign Labels Let x = hours flying Let y = distance in miles

Think of the point on a graph that represents 3hrs and 825 mi.

hours

distance

(3,825)

Use point-slope to write an equation.

11 xxmyy 3x275825y

825x275825y 825 825

x275y

11

y,x

bmxy b3752258

b825258 825 825

b0 x275y

Find the slope.

Assign Labels Let x = # lessons Let y = # vocab. words

Write an equation.

11 xxmyy

10x440y 40x440y

40 40 x4y

20,5

12

12

xx

yym

10 5

40 20

5

20

4

40,10

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.

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. Let x = Fri. hours Let y = Sat. hours

Verbal model. Friday’s earn. + Saturday’s earn. = total earn.

Write the equation. 36y6x4

CyBxA

Assign earning labels. Let 4x = Fri. earn. Let 6y = Sat. earn.

18y3x2

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.

x

y

•

Friday hours

Sat. hours

(0,6)

18y3x2 Make a table.

x y

0 6

09

•(9,0)•

•6 2

3 4 (6,2)

(3,4)

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

Verbal model. Cost of CDs + Cost of Videos = total cost

Write the equation. 75y5x15

CyBxA

Assign cost labels. Let 15x = cost of CDs Let 5y = cost of Videos

15yx3

Make a table and graph.

x y

15yx3

# of CDs

# o

f V

ideos

x

y

0 15

05

•

1 12

•

2 9•

3 6•

4 3

The ordered pairs represent the possible combinations.(0,15), (5,0), (1,12), (2,9), (3,6), (4,3)

•

•

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.

Identify two points.

Assign Labels Let x = year Let y = # comic books

Use slope-intercept to write an equation.

bmxy 60x18y

10,240 and 60,0

10 0

402 60

10

180

18

12

12

xx

yym

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.

Find the slope.

Assign Labels Let x = # weeks Let y = # laps

Use point-slope to write an equation.

11 xxmyy

1x14y

1x4y 4 4

3xy

4,1Identify a point.

1

4-A13 Handout A13