3.6 Distance. 3.6 – Equations & Problem Solving Goals / “I can…” Define a variable in terms...
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Transcript of 3.6 Distance. 3.6 – Equations & Problem Solving Goals / “I can…” Define a variable in terms...
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3.6
Distance
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3.6 – Equations & Problem Solving
Goals / “I can…” Define a variable in terms of another
variable Model distance-rate-time problems
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FirstThingsFirst!!
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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
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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.
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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.
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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
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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
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3.6 – Equations & Problem Solving
Consecutive Integers are numbers that differ by 1.
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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
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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?
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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
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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
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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
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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
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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
=
=
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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?
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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
=
=
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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
=
=