Download Chapter 3 Now You Can Solve Problems instead of just creating them!

Chapter 3
Now You
Can Solve
Problems
instead of
just
creating
them!
Intro to Equations
• Equation
– Can be Numerical or Variable
– Has an equals sign or >, <.
• 9+3=12
• 3x-2=10
True or False
A true equation
x+8=13
If x = 5 then
5+8= 13 Note: this is true
True or False
• False Equation
– If 9-2y=49
– So if we substitute 6 in for y
– Then 9-2*6=49
– This is a lie!
Solutions
• A solution to an equation is a
number that make the equation true.
• For example:
• Is -2 a solution of 2x-5=x2-3
• Lets find out by subbing in -2
• 2*(-2)-5 = (-2)2-3
•
-4-5 = 4-3
•
1=1
More examples
• Is -4 a sol’n of 5x-2=6x+2
• 5x-2=6x+2
• 5(-4)-2 = 6(-4)+2
• -20 -2 = -24 + 2
•
-22= -22
• YES!
Even more examples
• Is -4 a sol’n of 4+5x = x2-2x
• 4+5x = x2-2x
• 4+5(-4)=(-4)2-2(-4)
• 4+(-20)=16-(-8)
• -16=24
• NO!
Give it a try
• Is (1/4) a solution to 5-4x=8x+2?
• Is 5 a solution of 10x-x2=3x-10
Answers to you try it
Try a Few Harder Ones
• Is -6 a solution of 4x+3=2x-9
• Yes
• Is (-2/3) a solution of 4-6x=9x+1
• No
• Is -5 a solution of x2=25
• Yes’m
Opposites
• Remember: solving algebraic
equations is all about opposites.
• i.e. do the opposite of the whatever
the mathematical operation is.
Solving Stuff
• What you want at the end of all your
work
• The variable to = a constant
• Like y=5
• What's the opposite of:
• Addition
• Subtraction
• Multiplication
• Division
• Exponents
• Square Roots
Square Roots
• Break it down
• Examples:
• Square roots
of 49, 18, 27
– You try:
• Square roots
of 44, 96, 45
Back to where we were
• First form
• X+a=b
• X+3=5
• Try to get simplify first (PEMDAS)
• Try to isolate the variable
• Do the opposite
• X+3 =5
-3 -3
•X
=2
Example
• Y+3/4=1/2
• -3/4 -3/4
•
Y = -1/4
• Check your answer
• Sub in what you found for Y into the
original equation
• Does -1/4 +3/4 = ½
• You Bet!
Things are what they
appear?
• 3=T+2.5
• It’s the same thing– get everything
away from the variable.
•
3=T+2.5
• -2.5
-2.5
• 0.5=T
• Check your answer
Try These
•5=x+5
• x=0
• X-(1/4) = 5/6
• X= 13/12
The second type
• Form ax=b
• 2x=6
• What’s the operation between the 2 and
the x?
• What's the opposite?
• Do it!
• 2x=6
• 2 2
• x=3
More examples
• x/4=-9
• Division
• So Multiply
• (x/4)*4 = -9 *4
• X=-36
Tricky Problems
•
•
•
•
•
•
•
•
•
Ex1: 3x/4=5
For fractions, Multiply by the reciprocal!
In this case, multiply by 4/3
(4/3) *(3x/4) = 5*(4/3)
X=20/3
Ex2: 5x-9x=12
-4x=12
divide by -4
X=-3
You try it
• -2x/5 = 6
• -15
• 4x-8x = 16
• -4
• 8 = (3/4)x
• 32/3
• 2z = 0
• 0
Percent Problems
• Basic format
• Percent * Base = Amount
• Figure out which 2 they are giving
you.
• Key words
– Of means multiply
– Is means equals
Examples
• 20% of what number is 30?
• You are given the Percent and the
amount
• 20%*B=30
• 20% must be changed to a decimal
• 0.20*B=30
• Divide by 0.2
• B=150
Point of Interest?
Ex: During a recent year, nearly 1.2 million dogs or litters
were registered with the AKC?!. The lab retriever was the
most popular with 172,841 registered. What percent of the
registrations were labs? Round to the nearest tenth of a
percent. PS- Dogs are considered food in some
southeasters Asian countries. I heard labs are the tastiest.
What's given? B and A not P
P*(1,200,000)=172,841
Divide by 1,200,000
P = 0.144
Change to a percent = 14.4%
You try it
• 18 is 16.333% of what number?
• 108
• A telephone bill of $27.25 dollars consisted on charges for a
flat rate service, direct-dialed calls, and “other.” Of the
total, $3.27 was for direct-dialed calls. What percent of the
telephone bill was due to direct-dialed calls? What is a
direct-dialed call?
• 12%
• The total revenue for all football bowl games in 2000 was
about $158.3 million. The Big Ten conference got $22.45
million. What percent did it get?
• 14.2
Usury
• How to use unfamiliar formulas like:
• Simply Interest
• I=prt
– I = interest
– P=principal (not principle)
– r= simple interest rate
– T = time (in same units as rate!!!!)
Interesting Example (HA! HA!)
• Last month, Nirzwan paid $545 for a
Luv-Sac and had to use his credit
card. Yesterday, he got his monthly
bill and had to pay $8.72 in interest.
What is the annual interest rate on
the card?
• I=prt I = 8.72, p=545, t=1/12
• Solve and get r =0.192
Uniform Motion
• Factoid: When an object is in
uniform motion, the speed and
direction do not change.
• Uniform Motion Equation: d=rt
where d = distance, r = rate, t =
time.
Suppose…
• A car travels at 75 mph for 2 hours.
How far does it go?
• r=75mph, t = 2 hrs
• d = 75*2 = 150 miles.
Rate
Rate is distance divided by time
Best example: mph  Miles per hour
Could be anything: meters per
minute, inches per year, yards per
second, etc.
If James jogs four miles in thirty
minutes what is his jogging rate in
mph?
4 divided by 30 won’t do. The 30
minutes must be changed to hours by
divided by 60. Now, t = 30/60 = 0.5.
Rate = 4/0.5 = 8 mph. Not bad
considering he runs like a duck.
Examples
• Ted leaves his house at 8am and gets to
work at 8:30 am. He lives 15 miles away.
What is Ted’s speed?
• 30 mph
• Joan leaves her house and travels at an
average speed of 45 kph toward her shack
in the mountains 180 kilometers away.
How long will it take her to get to the
shack if she stops for a one hour lunch
break?
• 5 hours.
Try it before you buy it
• A plane that normally flies at 250 mph in
calm air (no ducks) is flying into a
headwind of 25 mph. How far can the
plane fly in 3 hours.
• 675 mi
• Two cars start from the same point and
move in opposite directions. One goes
west at 45 mph, and the other goes east
at 60 mph. In how many hours will the
cars be 210 mi apart. Hint combine rates!
• 2 hours
3.1 Homework
• 1 thru 154 EOO
• 163 thru 172 EOO
• 14, 22, 48, 74, 96, 104, 124,144,
164, 180.