1. Jon and Sara are planting tulip bulbs. Jon has planted 60 bulbs and is planting at a rate of 44 bulbs per hour. Sara has planted 96 bulbs and is planting at a rate of 32 bulbs per hour. In how many hours will Jon and Sara have planted the same number of bulbs? How many bulbs will that be?

2. Four times Greg's age, decreased by 3 is equal to 3 times Greg's age increased by 7. How old is Greg?

Greg is 10 years old.

After 3 hours, Jon and Sara will each have planted 192 bulbs.

WARM-UP

For systems of equations, slope intercept form will be used with word problems that:Have a specific starting

point and one rate of change

Have two different options that are to be set “equal” to each other

COPY

Consumer Economics Application

Jenna is deciding between two cell-phone plans. The first plan has a $50 sign-up fee and costs $20 per month. The second plan has a $30 sign-up fee and costs $25 per month. After how many months will the total costs be the same? What will the costs be? If Jenna has to sign a one-year contract, which plan will be cheaper? Explain.

Write an equation for each option. Let t represent the total amount paid and m represent the number of months.

Continued

Total paid is

sign-up fee plus

paymentamount

for eachmonth.

Option 1 t = $50 + $20 m

Option 2 t = $30 + $25 m

Step 1 t = 50 + 20mt = 30 + 25m

Both equations are solved for t.

Step 2 50 + 20m =30 + 25mSubstitute 50 + 20m for t in the second equation.

Step 3 50 + 20m = 30 + 25m Solve for m. Subtract 20m from both sides.–20m – 20m

50 = 30 + 5m Subtract 30 from both sides. –30 –30

20 = 5m Divide both sides by 5.

Write one of the original equations.

Step 4 t = 30 + 25m

t = 30 + 25(4)

t = 30 + 100

t = 130

Substitute 4 for m.Simplify.

Continued

5 5

m = 4

20 = 5m

Step 5 (4, 130)Write the solution as an

ordered pair.

In 4 months, the total cost for each option would be the same $130.

Jenna should choose the first plan because it costs $290 for the year and the second plan costs $330.

Continued

Option 1: t = 50 + 20(12) = 290

Option 2: t = 30 + 25(12) = 330

If Jenna has to sign a one-year contract, which plan will be cheaper? Explain.

Try This!

One cable television provider has a $60 setup fee and $80 per month, and the second has a $160 equipment fee and $70 per month.

a. In how many months will the cost be the same? What will that cost be.

Write an equation for each option. Let t represent the total amount paid and m represent the number of months.

Total paid is fee plus

paymentamount

for eachmonth.

Option 1 t = $60 + $80 m

Option 2 t = $160 + $70 m

Try This! Continued

Step 1 t = 60 + 80mt = 160 + 70m

Both equations are solved for t.

Step 2 60 + 80m = 160 + 70mSubstitute 60 + 80m for t in the second equation.

Step 3 60 + 80m = 160 + 70m Solve for m. Subtract 70m from both sides. –70m –70m

60 + 10m = 160 Subtract 60 from both sides.

Divide both sides by 10.

–60 –6010m = 10010 10

m = 10

Write one of the original equations.

Step 4 t = 160 + 70m

t = 160 + 70(10)

t = 160 + 700

t = 860

Substitute 10 for m.Simplify.

Try This! Continued

Tika and Candy decide to save money. Tika has $10 and will save $2 per week. Candy only has $5 but will save $3 per week. When will they have the same amount of money?

Candy’s Money Tika’s Money= = y2y1=

$5 plus $3 per week = $10 plus $2 per week

Identify the Variables!!!

Since the output is their total money that will be the “y” variable. Since the number of weeks is the same for both people and time marches on “x” variable is the number of weeks

y1 = y2

Substitute the “x” expressions 3x + 5 = 2x + 10 Use additive inverse to move the

variable x + 5 = 10 Use additive inverse to move the

constant x = 5 WEEKS Substitute back for y 3(5) + 5 = 20 dollars

Ray Allen and Rajon Rondo play for the Boston Celtics. They are both GREAT shooting guards. When the statistician totaled their baskets made, he found they had combined for 48 baskets. He did notice though that Ray had made twice as many baskets as Rajon. How many baskets did each player make?

Rondo’s Baskets

Rondo’s Baskets

+ =

= ●

Ray’s Baskets

Ray’s Baskets

Total

Baskets

TWO

If y = 2x REPLACE the y in this equation with 2x

Solve for x

Since x = 16 SUBSTITUTE THE QUANTITY

Solve for y

Rajon made 16 baskets

Ray made 32 baskets

Standard Form is used when you have TWO totals that relate two different groups. The objects will have two different rates!

HINT: This is ALWAYS the start of a system that involves total objects that create a total money! AND USUALLY begins with x + y = total objects!

COPY

Application

Paige has $7.75 to buy 12 sheets of felt and card stock for her scrapbook. The felt costs $0.50 per sheet, and the card stock costs $0.75 per sheet. How many sheets of each can Paige buy?

Write a system. Use f for the number of felt sheets and c for the number of card stock sheets.

0.50f + 0.75c = 7.75 The cost of felt and card stock totals $7.75.

f + c = 12 The total number of sheets is 12.

Continued

Step 1 0.50f + 0.75c = 7.75

+ (–0.50)(f + c) = 12

Multiply the second equation by –0.50 to get opposite f-coefficients.

0.50f + 0.75c = 7.75+ (–0.50f – 0.50c = –6)

Add this equation to the first equation to eliminate the f-term.

Simplify and solve for c.Step 2 0.25c = 1.75

c = 7

Step 3 f + c = 12

Substitute 7 for c.f + 7 = 12

–7 –7f = 5

Subtract 7 from both sides.

Write one of the original equations.

Write the solution as an ordered pair.

Step 4 (7, 5)

Paige can buy 7 sheets of card stock and 5 sheets of felt.

Continued

Step 5 (10, 860) Write the solution as an ordered pair.

In 10 months, the total cost for each option would be the same, $860.

The first option is cheaper for the first six months.

Continued

Option 1: t = 60 + 80(6) = 540

Option 2: t = 160 + 270(6) = 580

b. If you plan to move in 6 months, which is the cheaper option? Explain.

Application

What if…? Sally spent $14.85 to buy 13 flowers. She bought lilies, which cost $1.25 each, and tulips, which cost $0.90 each. How many of each flower did Sally buy?

Write a system. Use l for the number of lilies and t for the number of tulips.

1.25l + 0.90t = 14.85 The cost of lilies and tulips totals $14.85.

l + t = 13 The total number of flowers is 13.

Application Continued

Step 1 1.25l + .90t = 14.85+ (–.90)(l + t) = 13

Multiply the second equation by –0.90 to get opposite t-coefficients.

1.25l + 0.90t = 14.85 + (–0.90l – 0.90t = –11.70) Add this equation to

the first equation to eliminate the t-term.

Simplify and solve for l.Step 2

0.35l = 3.15

l = 9

Application Continued

Write the solution as an ordered pair.

Step 4 (9, 4)

Sally bought 9 lilies and 4 tulips.

Step 3 Write one of the original equations.Substitute 9 for l.9 + t = 13

–9 –9t = 4

Subtract 9 from both sides.

l + t = 13

Mike Bibby and Joe Johnson play for the Atlanta Hawks. Their combined points are 60. But the stats man noticed that if you subtracted two times Johnson’s points from three times Bibby’s points you would also have 60 points. Just how many did each of these Hawk’s score?

Verbal Models

Bibbys

Points+ =

Johnsons

Points

Total

Points

Three times Bibby

Points

- =Two times

Johnson

Points

Thirty

Points

x + y = 60 3x -2y = 60Think 2y would eliminate -2y so you need to distribute 2 to each term

2(x + y = 60)

2x + 2y = 120

Combine the Equations

2x + 2y = 120

5x = 1805 = 5

x = 36Now substitute the value into the first equation 36 + y = 60 y = 24

x = 36

y = 24

This means Bibby scored 36 points

This means Johnson scored 24 points

You are selling tickets to your high school play. Student tickets cost $4 and general admission cost $6. You sell 525 tickets and raise $2876. How many of each type of ticket did you sell?

# of Student Tkts

# of GenAdm Tkts

Total Tickets

525

=

x y

+

=

+

# of student tkts

# of Gen Adm tkts

Total Money

$4 per ticket

$6 per ticket

+

=● ●

28764x 6y =+

x + y = 525

Solve the equation for x

x = 525 – y

Now substitute the known value for y to find the student tickets

x = 525 – 388

x = 137 student tickets

4x + 6y = 2876

Substitute the “x” expression for x

4(525 – y) + 6y =2876

Distributive property

2100 – 4y + 6y = 2876

Combine like terms

2100 +2y = 2876

Solve for y

2y = 776

y = 388 gen adm tkts

Mike has one dollar bills and five dollar bills in his pocket. He has a total of 16 bills and knows he has $52. How many of each bill does he have?

# of one dollar bills

# of five dollar bills

Total number of bills

=

=

+

+# of one dollar bills

# of five dollar bills

●●RATE

$1

RATE

$5

x

1x

y

5y

16

52

+ =

+=

Total Amount of Money

You can solve the linear system by elimination method.

Lesson QuizSolve each system .

1. Harlan has $44 to buy 7 pairs of socks. Athletic socks cost $5 per pair. Dress socks cost $8 per pair. How many pairs of each can Harlan buy?

4 pairs of athletic socks and 3 pairs of dress socks

4 months13 stamps

2. Joy has 5 collectable stamps and will buy 2 more each month. Ronald has 25 collectable stamps and will sell 3 each month. After how many months will they have the same number of stamps? How many will that be?3. Plumber A charges $60 an hour. Plumber B charges $40 to visit your home plus $55 for each hour. For how many hours will the total cost for each plumber be the same? How much will that cost be? If a customer thinks they will need a plumber for 5 hours, which plumber should the customer hire? Explain. 8 hours; $480; plumber A: plumber A is cheaper for

less than 8 hours.