3.6

Distance

3.6 – Equations & Problem Solving

Goals / “I can…” Define a variable in terms of another

variable Model distance-rate-time problems

FirstThingsFirst!!

1) Solve 2x - 4y = 7 for xTo get x by itself, what is the first step?

1. Add 2x

2. Subtract 2x

3. Add 4y

4. Subtract 4y

Answer Now

Ask yourself, What is the first thing

we are doing to x? What is the second

thing?

1) Solve 2x - 4y = 7 for x Use a DO-UNDO chart to help determine the steps

DO UNDO

· 2

-4y

Follow the steps in the ‘undo’ column to isolate the variable.

+4y

÷ 2

Complete the undo column by writing the opposite operations in opposite order.

1) Solve 2x - 4y = 7 for x

1. Draw “the river”

2. Add 4y to both sides

3. Simplify

4. Divide both sides by 2

5. Does it simplify?

D U· 2 -4y

+4y ÷ 2

+ 4y + 4y

2x = 7 + 4y

2 27 4

2

yx

This fraction cannot be simplified because both terms in the numerator

are not divisible by 2.

3) Solve for y.

What is the first step?

y a3

c

1. Multiply by 3

2. Divide by 3

3. Add a

4. Subtract a

Answer Now

3) Solve for y:y a

3c

1. Draw “the river”

2. Clear the fraction – multiply both sides by 3

3. Simplify

4. Subtract a from both sides

5. Simplify

D U+ a ÷ 3

· 3 - a

y + a = 3c

-a -a

y = 3c - a

3 33

y ac

3.6 – Equations & Problem Solving

Consecutive Integers are numbers that differ by 1.

3.6 – Equations & Problem Solving

Example 1: The sum of three consecutive numbers is

72. Find them. The three numbers are x, x + 1, x + 2.

(x) + (x + 1) + (x + 2) = 72

3.6 – Equations & Problem Solving

Distance – Rate – Time Problems One of the most common and powerful

formulas in math and science is d = rt. This stands for

distance = rate x time. There are three types of uniform motion

problems: same direction, different direction, round trip.

HINT: How are the distances related?

The 3 formulas for Speed, Time & Distance:

Speed = Distance

TimeTime =

Distance

SpeedDistance =Speed x Time

Remember them from

this triangle:

D

S T

Solving for Speed Solving for Time Solving for Distance

D

S T

A windsurfer travelled 28 km in 1 hour 45 mins.

Calculate his speed.

Speed =

DistanceTime

28

1•75=

= 16 km/h

1 hour 45 mins

Answer: His speed was 16 km / Answer: His speed was 16 km / hourhour

2 hour 30 mins

Answer: He travelled 125 kmAnswer: He travelled 125 km

A salesman travelled at an average speed of 50 km/h for 2 hours 30 mins. How far did he travel?

D

S TDistance = Speed x Time

= 50 x 2•5

= 125 km

Answer: It took 9 hours 15 minutesAnswer: It took 9 hours 15 minutes

A train travelled 555 miles at an average speed of 60 mph. How long did the journey take?

D

S TTime =

DistanceSpeed

55560=

= 9•25 hours

= 9 hours 15 mins

3.6 – Equations & Problem Solving

Example 3: Same – Direction (SAME DIRECTION) A train leaves a train stations at 1 p.m. It travels at

a rate of 72 mi/hr. Another train leaves the same station at one hour later. It is traveling at 90 mi/hr. The second train follows the same path as the first on a parallel track. How long will it take for the second train to catch the first?

  rate Time

=

Distance

     =

 

      =  

3.6 – Equations & Problem Solving

A group of campers and their group leader left their campsite in a canoe. They traveled at 10 mi/hr. 2 hours later another group leader the same site in a motorboat. He traveled at 22 mi/hr. How long after the canoe left the site did

the motorboat catch the canoe? How long did the motorboat travel?

3.6 – Equations & Problem Solving

Example 4 Round Trip (SAME DISTANCE) You drive into town to get a new computer.

Because of traffic, you drive at 15 mi/hr. On your way home you drive 35 mi/hr. Your total trip is 2 hours. How long did it take you to get to the store?

  rate Time

=

Distance

      =  

      =  

3.6 – Equations & Problem Solving

Example 5 Opposite Direction (TOTAL DISTANCE) Jack and Jill leave their home in opposite

directions on the same road. Jack drives 15 mi/hr. faster than Jill. After 3 hours they are 225 miles apart. Find Jack’s rate and Jill’s rate.

  rate Time

=

Distance

      =  

      =