aim: how do we solve verbal problems using two variables?

Aim: Verbal Systems Course: Math Literacy

Jonathan left his home by car, traveling on a certain road at the rate of 45 mph. Three hours later, his brother Jessie left the home and started after him on the same road, traveling at a rate of 60 mph. In how many hours did Jessie overtake Jonathan?

Do Now:

Aim: How do we solve verbal problems using two variables?


Do Now

Jonathan traveled 3 hours longer

h = the number of hours Jessie traveled.

h + 3 = hours traveled by Jonathan

60h = the distance traveled by Jessie

60h

45(h + 3)

60h = 45(h + 3)60h = 45h + 13515h = 135

Jonathan

Jessie

45(h + 3) = the distance traveled by Jonathan

h = 9

60h = 45(h + 3)60(9) = 45(9 + 3)540 = 540

D = rt


Using Two Variables

The larger of two numbers is 4 times the smaller. If the larger number exceeds the smaller by 15, find the number.

Let x = smaller # Let 4x = larger #4x = x + 15 3x = 15 x = 5 4x = 20

Use a system of equations to solve the same problem

Use a system of equations to solve the same problem

x = y - 15

Let x = smaller # Let y = larger #y = 4x

Substitutionx = 4x - 15-3x = -15

x = 5 y = 4x = 4(5) = 20


Model Problem

The sum of two numbers is 8.6. Three times the larger number decreased by twice the smaller is 6.3. What are the numbers?

3y – 2x = 6.3

Let x = smaller # Let y = larger #

x + y = 8.6

2(x + y = 8.6) 2x + 2y = 17.2

-2x + 3y = 6.3

Additive inverse

x + y = 8.6

3y – 2x = 6.35y = 23.5

y = 4.7x + y = 8.6x + 4.7 = 8.6

x = 3.9

The two numbers are 4.7 and 3.9

Use a system of equations to solveUse a system of equations to solve


Do Now – Two Variables

Jonathan left his home by car, traveling on a certain road at the rate of 45 mph. Three hours later, his brother Jessie left the home and started after him on the same road, traveling at a rate of 60 mph. In how many hours did Jessie overtake Jonathan?

h = the number of hours Jessie traveled.

t = hours traveled by Jonathan

60h = the distance traveled by Jessie

60h = 45t

60h = 45(h + 3)15h = 135

45t = the distance traveled by Jonathan

h = 9

h + 3 = t


Mario had $6.50, consisting of dimes and quarters, in a coin bank. The number of quarters was 10 less than twice the number of dimes. How many coins of each kind did he have?

Do Now:

Aim: How do we solve verbal problems using two variables?


Use a system of equations to solve the same problem.

Let d = # of dimes q = # of quarters.10d = value of dimes

.25q = value of quarters

.10d + .25q = 6.50

q = 2d – 10 = 2(15) – 10 = 20

15 dimes = $1.50 20 quarters = $5.00

1.50 + 5.00 = $6.50

q = 2d - 10

10d + 25q = 650

Substitution

10d + 25(2d – 10) = 65010d + 50d – 250 = 650

60d – 250 = 65060d = 900

d = 15


Model Problem

6(6) + 8h = 14036 + 8h = 140 8h = 104h = 13

9b + 6h = 132

Let b = belt Let h = hat

The owner of a men’s clothing store bought six belts and eight hats for $140. A week later, at the same prices, he bought nine belts and six hats for $132. Find the price of a belt and the price of a hat.

6b + 8h = 140 Additive inverse - eliminate h

The belts costs $6 ea. And the hats cost $13 ea.

6b + 8h = 140

9b + 6h = 132

3(6b + 8h = 140)

-4(9b + 6h = 132)

18b + 24h = 420-36b - 24h = -528

-18b = -108b = 6


Let x = #lb. of cookie 1

Let y = #lb. of cookie 2

Use a system of equations to solveUse a system of equations to solveA dealer wishes to obtain 50 pounds of mixed cookies to sell for $3.00 per pound. If he mixes cookies worth $3.60 per pound with cookies worth $2.10 per pound, find the number of pounds of each kind he should use.

Value of cookie #1

Substitution

Value of cookie #23.60x 2.10yx + y = 50

3.60x + 2.10y = 150x = 50 - y

360x + 210y = 15000360(50 – y) + 210y = 1500018000 – 360y + 210y = 15000

– 150y = -3000y = 20 lb. of cookie 1 - $2.10

x + y = 50 x + 20 = 50 x = 30 lb. of cookie 2 - $3.60


Let r = boat’s rate in still water

Let c = current’s rate

Use a system of equations to solveUse a system of equations to solveA motor boat can travel 60 miles downstream in 3 hours. It requires 5 hours to make the return trip against the current. Find the rate of the boat in still water and the rate of the current.

Additive inverse -

eliminate c

r + c =

2r = 32

r + c = 20 r - c = 12

r - c =

boat’s rate going downstream

boat’s rate going upstream

20mph

12mph

r = 16mph rate of boat in still water

r + c = 20 16 + c = 20 c = 4mph rate of current

aim: how do we solve verbal problems using two variables?

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