Chapter 3Chapter 3Now You Now You

Can Solve Can Solve Problems Problems instead of instead of

just just creating creating them!them!

Intro to EquationsIntro to Equations• Equation

– Can be Numerical or Variable – Has an equals sign or >, <.

• 9+3=12• 3x-2=10

True or FalseTrue or False

A true equationx+8=13

If x = 5 then5+8= 13 Note: this is true

True or FalseTrue 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!

SolutionsSolutions• 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 examplesMore 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 examplesEven 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 tryGive 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 itAnswers to you try it

Try a Few Harder OnesTry 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

OppositesOpposites• Remember: solving algebraic

equations is all about opposites.• i.e. do the opposite of the whatever

the mathematical operation is.

Solving StuffSolving 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 RootsSquare Roots• Break it down• Examples:

• Square roots of 49, 18, 27

– You try:• Square roots

of 44, 96, 45

Back to where we wereBack 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

ExampleExample• 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 Things are what they appear?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 TheseTry These• 5 = x + 5• x=0

• X-(1/4) = 5/6• X= 13/12

The second typeThe 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 examplesMore examples• x/4=-9• Division • So Multiply • (x/4)*4 = -9 *4 • X=-36

Tricky ProblemsTricky 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 itYou try it• -2x/5 = 6• -15

• 4x-8x = 16• -4

• 8 = (3/4)x• 32/3

• 2z = 0• 0

Percent ProblemsPercent Problems• Basic format• Percent * Base = Amount• Figure out which 2 they are giving

you. • Key words

– Of means multiply– Is means equals

ExamplesExamples• 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?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 PP*(1,200,000)=172,841Divide by 1,200,000P = 0.144 Change to a percent = 14.4%

You try itYou 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 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!!!!)

InterestInteresting Example (HA! HA!)ing 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 MotionUniform 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…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.

RateRateRate is distance divided by timeBest example: mph Miles per hourCould 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.

ExamplesExamples• 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 itTry 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 Homework3.1 Homework• 1 thru 154 EOO• 163 thru 172 EOO• 14, 22, 48, 74, 96, 104, 124,144,

164, 180.