3.6
Distance
3.6 – Equations & Problem Solving
 Goals / “I can…”


Define a variable in terms of another
variable
Model distance-rate-time problems
1) Solve 2x - 4y = 7 for x
To get x by itself, what is the first step?
1. Add 2x
2. Subtract 2x
3. Add 4y
4. Subtract 4y
Answer Now
1) Solve 2x - 4y = 7 for x
Use a DO-UNDO chart to help determine the steps
Ask yourself,
 What is the first thing
we are doing to x?
 What is the second
thing?
Complete the undo column
by writing the opposite
operations in opposite
order.
DO UNDO
· 2 +4y
-4y ÷ 2
Follow the steps in
the ‘undo’
column to isolate
the variable.
D U
1) Solve 2x - 4y = 7 for x
1.
2.
3.
4.
5.
Draw “the river”
Add 4y to both
sides
Simplify
Divide both sides
by 2
Does it simplify?
·2
-4y
+4y
÷2
+ 4y + 4y
2x = 7 + 4y
2
2
7  4y
x
2
This fraction cannot be simplified
because both terms in the numerator
are not divisible by 2.
ya
 c for y.
3) Solve
3
What is the first step?
Multiply by 3
2. Divide by 3
3. Add a
4. Subtract a
1.
Answer Now
D U
3) Solve for y:
1. Draw “the river”
2. Clear the fraction
– multiply both
sides by 3
3. Simplify
4. Subtract a from
both sides
5. Simplify
3
+a
ya
÷3
c
3
ya
c 3
3
y + a = 3c
-a -a
y = 3c - a
·3
-a
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:
Distance
Speed =
Time
Solving for Speed
Time =
Distance
Speed
Solving for Time
Distance = Speed x Time
Solving for Distance
D
Remember them from
this triangle:
S
T
A windsurfer travelled 28 km in 1 hour 45 mins.
Calculate his speed.
Speed =
=
Distance
Time
28
1•75
D
S
T
1 hour 45 mins
= 16 km/h
Answer: His speed was 16 km / hour
A salesman travelled at an average speed of 50 km/h
for 2 hours 30 mins. How far did he travel?
Distance = Speed x Time
= 50 x 2•5
D
S
T
2 hour 30 mins
= 125 km
Answer: He travelled 125 km
A train travelled 555 miles at an average speed
of 60 mph. How long did the journey take?
Distance
Time =
Speed
D
S
=
555
60
=
9•25 hours = 9 hours 15 mins
T
Answer: It took 9 hours 15 minutes
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
=
=