Download a * b

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!
The Commutative Laws
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.)
Evaluate xy and yx when x = 7 and y = -5.
Solution
We substitute 7 for x and -5 for y.
xy = 7(-5) = -35
yx = -5(7) = -35
The Associative Laws
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.)
Calculate and compare:
4 + (9 + 6) and (4 + 9) + 6.
4 + 15
19
13 + 6
19
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
For any numbers a, b, and c,
a(b – c) = ab - ac.
Multiply.
4(a + b).
4*a+4*b
4a + 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).
Factoring
To factor an expression is to find an
equivalent expression that is a product.
Factor.
a.6x – 12
b. 8x + 32y – 8
a. 6x – 12
6*x–6*2
6(x – 2)
b. 8x + 32y - 8
8*x+8*4*y–8
8(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 = 6x
If the variables are different or have a different exponent
then they cannot be combined
2x + 4y, cannot be combined
3m + 5m², cannot be combined
Examples:
3x + 2 – 6x + y
3x – 6x + 2 + y
-3x + 2 + y
2x² - 5 + 9x + 23 – 8x² +3x
2x² - 8x² + 9x + 3x – 5 + 23
-6x² + 12x + 18
6(-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 – 9
8x = -6
Now what?
8x = -6
Remember that 8x just means 8 * x, so we need to get rid
of the 8:
8x/8 = -6/8
x = -6/8 Don’t forget to reduce
x = -3/4
Always check you answer:
9 + 8x = 3
x = -3/4
9 + 8(-3/4) = 3
9 - 24/4 = 3
9–6=3
3=3
Our 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 = 36
9x = 36
9x/9 = 36/9
x=4
Remember to always check you answer:
5x + 4x = 36
5(4) + 4(4) = 36
20 + 16 = 36
36 = 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 + 12
7 – 2x = 3x + 22
Now we need to get both x’s on the same side and finish:
7 – 2x + 2x = 3x + 22 + 2x
7 = 5x + 22
7 – 22 = 5x + 22 -22
-15 = 5x
-15/5 = 5x/5
-3 = x
Always check the answer:
4x + 7 – 6x = 10 + 3x + 12
4(-3) + 7 – 6(-3) = 10 + 3(-3) + 12
-12 + 7 + 18 = 10 – 9 + 12
-5 + 18 = 1 + 12
13 = 13
Correct!
An Equation-Solving Procedure
1. 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.
Copyright © 2008 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley
Slide 8- 30
3 – 8(x + 6) = 4(x – 1) – 5
Oh 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) – 5
That’s right! Use the distributive property of multiplication:
3 – 8x - 48 = 4x – 4 – 5
Now 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/-12
x = -3
Check our answer:
3 – 8(x + 6) = 4(-3 – 1) – 5
3 – 8(-3 + 6) = 4(-3 – 1) – 5
3 – 8(3) = 4(-4) – 5
3 – 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 / t
d/t = r or r = d/t
See?
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 = x
15.58 = x
15 is 16% of what?
Amount = percent * base
15 = 16% * n
15 = .16 * n
15/.16 = .16/.16 * n
93.75 = n
Check answer:
15 = 16% * 93.75
15 = .16 * 93.75
15 = 15
27 is what percent of 36?
Amount = percent * base
27 = v * 36
27/36 = v * 36/36
.75 = v
75% = v
Check answer:
27 = 75% * 36
27 = .75 * 36
27 = 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 = 22
This 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/.12
n = 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.