Chapter 3: Algebra

Prof. Carolyn Dupee

July 3, 2012

INTERPRETING MATH EXPRESSIONSPhrase (p. 135) Mathematical Expression

Six more than a number. X + 6

A number increased by 3. X + 3

Four less than a number. X - 4

A number decreased by 9. X - 9

Twice a numer. 2x

3 decreased by a number. 3 - x

Four less than 3 times a number. 3x - 4

Ten more than twice a number 2x + 10

Eight times a number, decreased by 7.

8x -7

Six more than a number is ten. X + 6 = 10

Twice a number, decreased by 6 is 12.

2x – 6 =12

2

POLL QUESTIONS:

1. 8 more than z.

2. 4 more than 2 times x.

3. 5 decreased by 4 times c.

3

WRITE THE EQUATION AND SOLVE.

•5 multiplied by a number is 35. •5 n = 35•To solve, do the opposite process of what is happening to the variable (n). 5 is multiplied, so the opposite is division. Do to both sides of the equation

•5 n = 35•÷5 ÷5•n = 7

4

COMBINING LIKE TERMS

Like Terms Unlike Terms (You can’t combine!)

2x, 7 x (same variable x) 2x, 9 (2x= variable, 9= constant)

-8y, 3y (same variable y) 5x, 6y (not like, 2 different variables x and y)

-4, 10 (like, both constants) X, 8 (not like, variable and constant)

-5x2, 6x2 (like, same variable with same exponents)

2x3, 3x2 (not like, 2 different exponents)

5

COMBINING LIKE TERMS

Ex. 1) -3x + 8x -5

6

EVALUATING & USING FORMULAS

1. Plug the number in for the variable.

2. Solve the expression.-Ex. 1) Evaluate y2 for y=10-y2 = (10)2 = 100

7

EVALUATING & USING EQUATIONS

•Ex. 2) D = r t where r is $11 and t= 4 hours-(when 2 variables are next to each other it means to multiply).

-D= $11*4 = $44/ hour

8

EVALUATING & USING EQUATIONS

•F = 9/5 C + 32 (multiply the 9 * C) divide by 5 then add 32•C= 100oC, what is the temperature in Farenheit?•9 (100)/ 5 = 180 + 32 = 212 oF

9

EVALUATING & USING EQUATIONS

•Determine whether 12 is a solution of the equation, 4x + 12= 48.- Step 1: Subtract 12 from both sides of equation to get the variable on one side

and constant on the other side. 4x + 12-12 = 48 -12. 4x = 36- Step 2: Get the variable by itself, 4*x, do the opposite ÷ both sides by 4. 4x/4 =

36/4 so x=9.

10

EVALUATING & USING EQUATIONS

•A = P (1 + r t) where A = 2000, r= 0.05, t= 6)- Step 1: Substitute the numbers into the equation. 2000 = P (1 + (0.05)(6))- Step 2: Simplify the parentheses, using order of operations.

- (.05)(6) = 0.3 + 1= 1.03- Step 3: Rewrite the equation, the new number 1.03 is now multipled by P.

- 2000 = 1.03 P- Step 4: Solve for P, do the opposite what’s done with the P. Divide both sides by

1.03. - 2000/1.03 = 1.03 P/ 1.03 - 1941.747573 = P (Round to the nearest dollar) P= 1941.75

11

EVALUATING & USING EQUATIONS

•You will see this equation again in the next chapter!

• (y2-y1) / (x2-x1) where y2= -6, x2= 4, y1= -2, x1= 2- Step 1: Substitute in your numbers. (Be careful to use the proper order).

(-6 - - 2) / (4 – 2)

Step 2: Remember that if you have two negatives in a row, it equals a positive. (-6 + 2)/ (4-2)

Step 3: Combine terms on top, and terms on bottom. -4/2

Step 4: Simplify your answer. -4/2 = -2

If your answer is a fraction, you can leave it as a fraction.

12

SOLVING MULTI-STEP EQUATIONS

•8x + 8y = 3, Solve for y- Step 1:Get the y on one side by subtracting 8x from both sides.• 8x -8x + 8y = -8x + 3• 8y = -8x + 3• Step 2: Divide both sides by the number multiplied by the variable (÷8)• 8y/8 = -8x/8 + 3/8• y = -x + 3/8 (you can leave this as a fraction).

13

SOLVING MULTI-STEP EQUATIONS

Poll: Solve for c: b = 4a + 5c-What’s the first step?-What’s the second step?-What’s the answer to the equation?

-(You can have multiply correct responses to this one base on notation).

14

USING THE DISTRIBUTIVE PROPERTY

Ex. 1) 4 (4x + 3) = 12 – (x – 3)Step 1: Multiply each term by the number outside the parentheses.

16 x + 12 = 12 – (x – 3)

Step 2: Assume the negative outside the second part of the parentheses is a -1 and multiply all terms inside the parentheses.

16 x + 12 = 12 –x + 3

Step 3: Combine like terms one one side.

16x + 12 = 15 – x

Step 4: Combine like terms on each side of equation.

15

USING THE DISTRIBUTIVE PROPERTY

Step 4: Combine like terms on each side of equation with variables.

16x + 12 = 15 – x

+x +x

Step 5: combine like terms on other side of equation with constants.

17 x +12 = 15

-12 -12

Step 6: Solve for x. 17 x/ 17 = 3/ 17

X= 3/17

16

SOLVING EQUATIONS WITH PROPORTIONS

• If it costs $1.50 for 1000 gallons of water, how much would it cost for 1350 gallons of water? (Use the whiteboard)

- Step 1: Set up the proportion (be sure the tops and bottoms are the unit)- $1.50/ 1000 gallons = x / 1350 gallons

- Step 2: Cross multiply. When you do so, you multiply diagonally across the equal signs.

- (1.50)(1350) = (1000)(x)- 2025 = 1000x

- Solve for x. - 2025/1000 = 1000x/1000 x= $2.025 round to $2.03

17

SOLVING WORD PROBLEMS

1. Write down what you know and what you’re trying to find.

2. Draw a picture if possible.

3. Pick a letter for what you’re trying to find.

4. Write the word problem as an equation.

5. Solve the equation for the unknown quantity.

6. Answer the question.

7. Double check that the answer makes logical sense!

18

WORD PROBLEMS

•P. 140 Ex. 48: Hollywood Videos offers customers two options to rent DVDs. The Unlimited Option costs $28 a month for unlimited DVDs. The Single Option costs $3.50 per DVD rental. How many DVDs must be rented for the two options to be equal?

- $28 = 3.50 D- $28/3.50 = 3.50 D/ 3.50- 8 DVDs

19

P. 139, EX. 31

•When sales representatives for Pfizer Pharmaceuticals drive to out-of-town meetings that require an overnight stay, they receive a $150 for lodging plus $0.42 per mile driven. How many miles did Joe Kotaska drive if Pfizer reimbursed him $207.54 for an overnight trip.o$150 + $0.42 m = $207.54o-$150 -$150o$0.42 m = $57.54o$0.42 m/$0.42 = $57.54/$0.42 = 137 miles

20

P. 133 EX. 71 SAVINGS ACCOUNT

•Christine Northrup borrowed $4500 from a bank at a simple interest rate of 2.5% for one year.

-A) Determine how much interest Christine paid at the end of 1 year.-2.5% convert it to a decimal move over 2 decimal places 0.025-$4500* 0.025 = $112.5-B) Determine the total amount Christine will repay the bank at the end of 1 year.

-$4500 + $112.50 = $4612.50

21

P. 139 EX. 34 MODELING- PET SUPPLIES

•PetSmart has a sale offering 10% off of all pet supplies. If Amanda Miller spent $15.72 on pet supplies before tax, what was the price of the pet supplies purchased before the discount?

•Step 1: Write the equation. 1x- 0.10x = $15.72•Step 2: Simplify by multiplying by 100. 100 (x-0.10x) = 100($15.72)•Step 3: Multiply. 100x – 10x = $1572 90x = $1572•Step 4: Solve for x. 90x/90 = $1572/90 x= $17.466666667 •X = $17.47

22