1. Chapter 3 Solving Equations
Introduction to Equations
Equation: equality of two mathematical expressions. =
9 + 3 = 12
3x – 2 = 10
y² + 4 = 2y - 1

2. Solution to an equation, is the number
when substituted for the variable makes the equation a true statement.
Is –2 a solution or 2x + 5 = x² - 3 ?
Substitute –2 in for the x
2(-2) + 5 = (-2)² - 3
= 4 - 3
1 = 1

3. Solve an equation Addition Property
r – 6 = We use the Addition
method by adding
positive 6 to both sides
of the equation.
r = 20
*CHECK your solution

4. Solve an equation
s + ¾ = ½
- ¾ -¾ Using the Addition
Method add a negative
¾ to both sides.
s = -¼ Remember to get a common
denominator.
Check your solution.

5. Solving Equations
3y = 27
Using the Multiplication
Method we divide by the coefficient, which is the same as multiplying by ⅓
3y = 27
y = 9
Check your solution

6. Solving Equations
Using the multiplication
method we multiply the
reciprocal of the
coefficient to both sides.
X = 10
Check 4/5(10) = 8
8 = 8

7. Solving Equations: 2 Step
6x + 12 = 36
6x + 12 = 36
Addition Method
6x = 24
6x = 24
Multiplication
Method
x = 4
Check

8. Basic Percent Equations Percent • Base = Amount P • B = A
20% of what number is 30
multiply B
equals
.2 • B = 30
B = 150

9. Basic Percent Equations Percent • Base = Amount P • B = A
What Percent of 80 is 70
P multiply equals
P • 80 = 70
P = .875
P = 87.5%
Convert to percentage.

10. Basic Percent Equations Percent • Base = Amount P • B = A
25% of 60 is what?
multiply
equals
amount
.25 • 60 = A
15 = A

11. Steps to solve equations:
1. Remove all grouping symbols
Look to collect the left side and
the right side.
Add the opposite of the smallest
variable term to each side.
Add the opposite of the constant
that’s on the same side as the
variable term to each side.

12. Steps to solve equations continued
5. Divide by the coefficient.
*variable term = constant term
*if the coefficient is a fraction,
multiply by the reciprocal.
6. CHECK the solution.

13. Ex. Solving Equations
3x – 4(2 – x) = 3(x – 2) - 4
3x – 8 + 4x = 3x – 6 – 4
Distribute
7x – 8 = 3x - 10
Collect like terms
-3x x
4x – 8 = -10
Add opposite of the
Smallest variable term
4x = -2
Add the opposite of
the constant
Divide by the
Coefficient.
x = -½

14. Ex. 2 Solving Equations
-2[4 – (3b + 2)] = 5 – 2(3b + 6)
-2[4 – 3b – 2] = 5 – 6b - 12
-8 + 6b + 4 = 5 – 6b - 12
6b – 4 = -6b - 7
Collected
12b – 4 = -7
Added 6b
12b = -3
Added 4
b = ¼
Divided by 12 and reduced
CHECK

15. Translating Sentences into Equations
Equation-equality of two mathematical
expressions.
Key words that mean =
equals is
is equal to
amounts to
represents

16. Ex. Translate:
“five less than a number is thirteen”
n - 5 = 13
n = 18
Solve

17. Translate Consecutive Integers
Consecutive integers are integers that
follow one another in order.
Consecutive odd integers- 5,7,9
Consecutive even integers- 8,10,12

18. CHAPTER 4 POLYNOMIALS
Polynomial: a variable expression
in which the terms are monomials.
Monomial: one term polynomial
5, 5x², ¾x, 6x²y³ Not: or 3
Binomial: two term polynomial
5x² + 7
Trinomial: Three term polynomial
3x² - 5x + 8

19. Addition and Subtraction
Polynomials can be added vertically
or horizontally.
Horizontal Format
Collect like terms
Ex. ( 3x³ - 7x + 2) + (7x² + 2x – 7)
3x³
+ 7x²
- 5x
- 5

20. Addition and Subtraction
Vertical Format
Ex. ( 3x³ - 7x + 2) + (7x² + 2x – 7)
³ ² ¹ º
3x³ x + 2
+7x² + 2x – 7
Organized in
columns by the
degree
3x³+7x² - 5x - 5

21. Subtraction
Horizontal Format
(-4w³ + 8w – 8) – (3w³ - 4w² - 2w – 1)
Change subtraction to addition of the opposite
(-4w³ + 8w – 8)+(-3w³ + 4w² + 2w + 1)
-7w³
+ 4w²
+ 10w
- 7

22. Subtraction
Vertical Format
(-4w³ + 8w – 8) – (3w³ - 4w² - 2w – 1)
Change
subtraction to
the addition of
the opposite
³ ² ¹ º
-4w³ w - 8
-3w³ + 4w² +2w + 1
-7w³ - 4w² + 6w - 9

23. Multiplication of Monomials
Remember x³ = x • x • x
& x² = x • x
Then x³ • x² = x • x • x • x • x = x5
RULE 1 xn • xm = x n+m
when multiplying similar
bases add the powers.
Ex. y4 • y • y3 = y = y8

24. Multiplying Monomials
Ex. (8m³n)(-3n5)
*Multiply the coefficients,
*Multiply similar bases by
adding the powers together
-24m3n6

25. Simplify powers of Monomials
(x4)3 = x4 • x4 • x4 = x = x12
Rule 2 (x m)n = xmn
Multiply the outside power
with the power on the inside.
Rule 3 (xmyn)p = xmpynp
Ex. (5x²y³)³ = 51•3x2•3y3•3 = 125x6y9

26. Simplify Monomials Continue
Ex.
(ab²)(-2a²b)³
Rule 3: Multiply
the outside power
to inside powers.
(ab²)(-8a6b³)
-8a7b5
Rule 1: multiply the
Monomials by adding
the exponents

27. Multiplication of Polynomials
Distribute and follow
Rule 1
-3a(4a² - 5a + 6)
-12a³
+ 15a²
- 18a

28. Multiplication of two Polynomials
*when multiplying two polynomials
you will use Distributive Property.
*be sure every term in one parenthesis
is multiplied to every term in the
other parenthesis.

29. Multiplication of two Polynomials
Ex.
(y – 2)(y² + 3y + 1)
Multiply y to
every term.
Multiply –2 to y³
+ 3y²
+ y
- 2y²
- 6y
- 2
Combine like
terms.
y³ + y² - 5y - 2

30. Multiply two Binomials
The product of two binomials can be
found using the FOIL method.
F First terms in each parenthesis.
O Outer terms in each parenthesis.
I Inner terms in each parenthesis.
L Last terms in each parenthesis.

31. Multiply two Binomials
Ex. (2x + 3)(x + 5) O F F O I L
(2x + 3)(x + 5) =
2x²
+10x
+3x
+15 I L
Collect Like Terms
2x² + 13x + 15

32. Special Products of Binomials
Sum and Difference of two Binomials
(a + b)(a – b)
Square the first term
Square the second
term a² - b²
Minus sign between the products

33. Sum and Difference of Binomials
(2x + 3)(2x – 3)
Square the term 2x -
4x² 9
Square the term 3
Minus sign between the terms

34. Or use the short cut Square of a Binomial (a + b)² = (a + b)(a + b)
Then FOIL
Or use the short cut
(a + b)² = a²
+ 2ab
+ b²
1. Square 1st term
2. Multiply terms
and times by 2.
ab times 2 =
3. Square 2nd term

35. Square of a Binomial
Ex. (5x + 3)² =
25x²
+ 30x
+ 9
Square 5x
Multiply 5x and 3
then times by 2
Square the 3

36. Square of a Binomial (4y – 7)² = 16y² - 56y + 49 Square the 4y
Mutiply the 4y and 7
then times by 2
Square the 7

37. Integer Exponents Divide Monomials
x•x•x•x•x
x•x x³ = =
Rule 4 xm xn
When m > n
Xm-n = xm xn 1
xn-m
When n > m =

38. Integer Exponents Divide Monomials
r8t6
r5t
r8-5t6-1
= r3t5 = 1
a6-4b9-7 1
a²b²
a4b7
a6b9 = =
a5b3c8d4
a2b7c4d9
a5-2c8-4
b7-3d9-4
a3c4
b4d5 = =

39. Integer Exponents Zero and Negative Exponents
Rule 5 a0 = a ≠ 0 x³
= x3-3
= xº
*Summary any number (except for 0) or variable raised to the power of zero = 1

40. Integer Exponents Zero and Negative Exponents
x-n = 1 xn
Rule 6: 1
X-n
= xn
and 1 1
If we make everything a fraction,
we can see that we take the base and it’s negative exponent and move them from the numerator to the denominator
and the sign of the exponent changes.

41. Integer Exponents Zero and Negative Exponents2
5a-4
= 2a4 5