More Algebra! Oh no!. Examples: 6 * 3 = 18, positive * positive = positive -8 * 5 = -40, negative * positive = negative -7.5 * -2.8 = 21, negative * negative.

More Algebra! Oh no!

Examples:

6 * 3 = 18, positive * positive = positive

-8 * 5 = -40, negative * positive = negative

-7.5 * -2.8 = 21, negative * negative = positive

Practice:

-12 * 5 =

-3.2 * -28 =

-3 * 5 * (-14) =

_4² =

-6³ =

Practice:

-12 * 5 = -60

-3.2 * -28 = 89.6

-3 * 5 * (-14) = 210

_4² = (-4 * -4) = 16

-6³ = (-6 * -6 * -6) = -216

Evaluate 3x2 when x = 4 and x = -4.

3(4)² 3(16) 48

3(-4)² 3(16) 48

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slide 7- 5

Same as multiplication! Duh!

Addition: For any numbers a, and b,a + b = b + a.

(We can change the order when adding without affecting the answer.)

Multiplication. For any numbers a and b,ab = ba

(We can change the order when multiplying without affecting the answer.)

The Commutative Laws

Evaluate xy and yx when x = 7 and y = -5.

SolutionWe substitute 7 for x and -5 for y.xy = 7(-5) = -35

yx = -5(7) = -35

Addition: For any numbers a, b, and c,a + (b + c) = (a + b) + c.

(Numbers can be grouped in any manner for addition.)Multiplication. For any numbers a, b, and c,

a * (b * c) = (a * b) * c(Numbers can be grouped in any manner for

multiplication.)

The Associative Laws

Calculate and compare:4 + (9 + 6) and (4 + 9) + 6.

4 + 15 13 + 619 19

For any numbers a, b, and c,a(b + c) = ab + ac.

The Distributive Law of Multiplication over Addition

For any numbers a, b, and c,a(b – c) = ab - ac.

The Distributive Law of Multiplication over Subtraction

Multiply.

4(a + b).

4 * a + 4 * b4a + 4b

Practice:Multiply. 1.8(a – b) 2.(b – 7)c 3.–5(x – 3y + 2z)

Practice:Multiply. 1.8(a – b) = 8a – 8b 2.(b – 7)c = bc – 7c 3.–5(x – 3y + 2z) = -5x + 15y – 10z

Factoring is the reverse of multiplying. To factor, we can use the distributive laws in reverse:ab + ac = a(b + c) and ab – ac = a(b – c).

To factor an expression is to find an equivalent expression that is a product.

Factoring

Factor.a. 6x – 12 b. 8x + 32y – 8

a. 6x – 126 * x – 6 * 26(x – 2)

b.8x + 32y - 88 * x + 8 * 4 * y – 88(x + 4y – 8)

Practice:Factor. a. 7x – 7y b. 14z – 12x – 20

Practice:Factor. a. 7x – 7y 7 * x – 7* y 7(x – y)

b. 14z – 12x – 20 2 * 7z – 2 * 6x – 2 * 10 2(7z – 6x – 10)

When the variable is exactly the same then the terms can be combined:

2x + 4x = 6xIf the variables are different or have a different

exponent then they cannot be combined2x + 4y, cannot be combined3m + 5m², cannot be combined

Examples:

3x + 2 – 6x + y3x – 6x + 2 + y-3x + 2 + y

2x² - 5 + 9x + 23 – 8x² +3x2x² - 8x² + 9x + 3x – 5 + 23-6x² + 12x + 186(-x² + 2x + 3)

Equation

An equation is a number sentence that says that the expressions on either side of the equals sign, =, represent the same number.

9 + 8x = 3

We need to isolate x to solve the equation. We have been doing this all term. What do we do first?

9 + 8x = 3

First, we get rid of the 9

9 + 8x - 9 = 3 – 98x = -6

Now what?

8x = -6

Remember that 8x just means 8 * x, so we need to get rid of the 8:

8x/8 = -6/8x = -6/8 Don’t forget to reducex = -3/4

Always check you answer:

9 + 8x = 3x = -3/49 + 8(-3/4) = 39 - 24/4 = 39 – 6 = 33 = 3Our answer is correct!

Let’s try another:5x + 4x = 36

Here we need to combine like terms. That means anything with the same variable next to it can be added together:

5x + 4x = 369x = 369x/9 = 36/9x = 4Remember to always check you answer:5x + 4x = 365(4) + 4(4) = 3620 + 16 = 3636 = 36

Here is a toughie!

4x + 7 – 6x = 10 +3x + 12

Holy cow! What do we do with this?!

4x + 7 – 6x = 10 +3x + 12

First, we combine like terms:4x + 7 – 6x = 10 + 3x + 127 – 2x = 3x + 22Now we need to get both x’s on the same side and

finish:7 – 2x + 2x = 3x + 22 + 2x7 = 5x + 227 – 22 = 5x + 22 -22-15 = 5x-15/5 = 5x/5-3 = x

Always check the answer:4x + 7 – 6x = 10 + 3x + 124(-3) + 7 – 6(-3) = 10 + 3(-3) + 12-12 + 7 + 18 = 10 – 9 + 12-5 + 18 = 1 + 1213 = 13

Correct!

Slide 8- 30Copyright © 2008 Pearson Education, Inc.

Publishing as Pearson Addison-Wesley

An Equation-Solving Procedure1. Multiply on both sides to clear the equation of

fractions or decimals. (This is optional, but can ease computations.) Not a fan of this one.

2. If parentheses occur, multiply to remove them using the distributive laws.

3. Collect like terms on each side, if necessary.4. Get all terms with variables on one side and all

numbers (constant terms) on the other side, using the addition principle.

5. Collect like terms again, if necessary. 6. Multiply or divide to solve for the variable, using the

multiplication principle.7. Check all possible solutions in the original equation.

3 – 8(x + 6) = 4(x – 1) – 5Oh no! Parentheses! Be afraid, be very afraid!

(Not really)

What do we do first to get rid of the parentheses?

3 – 8(x + 6) = 4(x – 1) – 5That’s right! Use the distributive property of

multiplication:3 – 8x - 48 = 4x – 4 – 5Now we just do the regular stuff:-8x - 45 = 4x – 9-8x - 45 -4x = 4x – 9 -4x-12x - 45 = -9-12x - 45 + 45 = -9 + 45-12x = 36-12x/-12 = 36/-12x = -3

Check our answer:3 – 8(x + 6) = 4(-3 – 1) – 53 – 8(-3 + 6) = 4(-3 – 1) – 53 – 8(3) = 4(-4) – 53 – 24 = -16 – 5-21 = -21

Got it!

Formulas are often composed of multiple variables. It is important to know how to change the equation around to solve for the unknown variable:

d = r * t, d is distance, r is rate, and t is time

Suppose we know the t and distance. Let’s change the equation to solve for r:

d/t = r * t / td/t = r or r = d/tSee?

To solve a problem involving percents, it is helpful to translate first to an equation. For example, “23% of 5 is what?”

23% of 5 is what? 0.23 * 5 = a

What is 19% of 82?

This is a reverse version of the previous example:

Amount = Percent number * Base or

Percent number * Base = Amount

19% * 82 = x.19 * 82 = x15.58 = x

15 is 16% of what?Amount = percent * base15 = 16% * n15 = .16 * n15/.16 = .16/.16 * n93.75 = n

Check answer:15 = 16% * 93.7515 = .16 * 93.7515 = 15

27 is what percent of 36?Amount = percent * base27 = v * 3627/36 = v * 36/36.75 = v75% = v

Check answer:27 = 75% * 3627 = .75 * 3627 = 27

Five Steps for Problem Solving in Algebra

1. Familiarize yourself with the problem situation.

2. Translate the problem to an equation.

3. Solve the equation.

4. Check the answer in the original problem.

5. State the answer to the problem clearly.

Digicon prints digital photos for $0.12 each plus $3.29 shipping and handling. Your weekly budget for the school yearbook is $22.00. How many prints can you have made if you have $22.00?

We first need to make an equation. Let’s use n for the number of prints we can make.

.12 * n is the cost per picture.We need to add 3.29 for shipping, so we have:(.12 * n) + 3.29 = cost of pictures

Digicon prints digital photos for $0.12 each plus $3.29 shipping and handling. Your weekly budget for the school yearbook is $22.00. How many prints can you have made if you have $22.00?

We know that we can only spend $22, so that is our total cost:

(.12 * n) + 3.29 = 22This is our equation.

Now that we have our equation, we just solve for our variable:

(.12 * n) + 3.29 = 22.12n + 3.29 – 3.29 = 22 -3.29.12n = 18.71.12n/.12 = 18.71/.12n = 155.916666

Remember that our answer needs to be in number of pictures, so the most pictures we can develop in one week is 155.