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Chapter 3 Solving Equations
Introduction to Equations
Equation: equality of two
mathematical expressions.
9 + 3 = 12
3x – 2 = 10
y² + 4 = 2y - 1
=
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 + 5 = 4 - 3
1=1
Solve an equation Addition Property
r – 6 = 14
r – 6 = 14 We use the Addition
+ 6 +6 method by adding
positive 6 to both sides
of the equation.
r = 20
*CHECK your solution
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.
Solving Equations
3y = 27 Using the Multiplication
Method
we
divide
by
the
3y = 27
coefficient,
which
is
the
3 3
same as multiplying by ⅓
y=9
Check your solution
Solving Equations
4
Using the multiplication
x8
5
method we multiply the
5 4
5 reciprocal of the
 x  8
4 5
4 coefficient to both sides.
X = 10
Check 4/5(10) = 8
8=8
Solving Equations: 2 Step
6x + 12 = 36
6x + 12 = 36
Addition Method
- 12 -12
6x = 24
6x = 24
6
6
x=4
Multiplication
Method
Check
Basic Percent Equations
Percent • Base = Amount
P • B =A
20% of what number is 30
multiply B
.2 • B = 30
B = 150
equals
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.
Basic Percent Equations
Percent • Base = Amount
P • B =A
25% of 60 is what?
multiply equals amount
.25 • 60 = A
15 = A
Steps to solve equations:
1. Remove all grouping symbols
2. Look to collect the left side and
the right side.
3. Add the opposite of the smallest
variable term to each side.
4. Add the opposite of the constant
that’s on the same side as the
variable term to each side.
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.
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
Add opposite of the
-3x
-3x
Smallest variable term
4x – 8 = -10
+8 +8
Add the opposite of
4x = -2
the constant
4 4
Divide by the
x = -½
Coefficient.
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
Added 4
12b = -3
Divided by 12
b=¼
CHECK
and reduced
Translating Sentences into Equations
Equation-equality of two mathematical
expressions.
Key words that mean =
equals
is
is equal to
amounts to
represents
Ex. Translate:
“five less than a number is thirteen”
n-5
Solve
= 13
n = 18
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
CHAPTER 4 POLYNOMIALS
Polynomial: a variable expression
in which the terms are monomials.
Monomial: one term polynomial
xy
5, 5x², ¾x, 6x²y³ Not: r or 3 x
Binomial: two term polynomial
5x² + 7
Trinomial: Three term polynomial
3x² - 5x + 8
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
Addition and Subtraction
Vertical Format
Ex. ( 3x³ - 7x + 2) + (7x² + 2x – 7)
³ ²
¹ º
Organized in
3x³
- 7x + 2
columns by the
+7x² + 2x – 7
degree
3x³+7x² - 5x - 5
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
Subtraction
Vertical Format
(-4w³ + 8w – 8) – (3w³ - 4w² - 2w – 1)
Change
³
²
¹ º
-4w³
+ 8w - 8 subtraction to
-3w³ + 4w² +2w + 1 the addition of
the opposite
-7w³ - 4w² + 6w - 9
Multiplication of Monomials
Remember x³ = x • x • x
& x² = x • x
Then x³ • x² = x • x • x • x • x = x5
•
x
when multiplying similar
bases add the powers.
Ex. y4 • y • y3 = y 4+1+3 = y8
RULE 1
n
x
m
x =
n+m
Multiplying Monomials
Ex. (8m³n)(-3n5)
*Multiply the coefficients,
*Multiply similar bases by
adding the powers together
3
6
-24m n
Simplify powers of Monomials
= • • =
Rule 2 (x m)n = xmn
Multiply the outside power
with the power on the inside.
4
3
(x )
4
x
4
x
4
x
4
+
4
+
4
x
=
12
x
Rule 3 (xmyn)p = xmpynp
Ex. (5x²y³)³ =
1•3
2•3
3•3
5 x y
=
6
9
125x y
Simplify Monomials Continue
Ex.
Rule 3: Multiply
(ab²)(-2a²b)³ the outside power
to inside powers.
(ab²)(-8a6b³)
7
5
-8a b
Rule 1: multiply the
Monomials by adding
the exponents
Multiplication of Polynomials
-3a(4a² - 5a + 6)
-12a³ + 15a² - 18a
Distribute
and follow
Rule 1
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.
Multiplication of two Polynomials
Ex.
(y – 2)(y² + 3y + 1) Multiply y to
every term.
Multiply –2 to
y³ + 3y² + y
every term.
- 2y² - 6y - 2
Combine like
y³ + y² - 5y - 2
terms.
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.
Multiply two Binomials
Ex. (2x + 3)(x + 5)
F O
F O I L
(2x + 3)(x + 5) = 2x² +10x +3x +15
I L
Collect Like Terms
2x² + 13x + 15
Special Products of Binomials
Sum and Difference of two Binomials
(a + b)(a – b) Square the first term
a² - b²
Square the second
term
Minus sign between
the products
Sum and Difference of Binomials
(2x + 3)(2x – 3)
Square the term 2x
4x² - 9
Square the term 3
Minus sign between
the terms
Square of a Binomial
(a + b)² = (a + b)(a + b) Then FOIL
Or use the short cut
(a + b)² = a² + 2ab + b²
ab times 2 =
1. Square 1st term
2. Multiply terms
and times by 2.
nd
3. Square 2 term
Square of a Binomial
Ex. (5x + 3)² = 25x² + 30x + 9
Square 5x
Multiply 5x and 3
then times by 2
Square the 3
Square of a Binomial
(4y – 7)² = 16y² - 56y + 49
Integer Exponents
Divide Monomials
x5 x•x•x•x•x x³
=
=
2
x
x•x
Rule 4
m
x
n
x
m-n When m > n
X
=
xm
1
=
xn xn-m
When n > m
Integer Exponents
Divide Monomials
8
6
8-5t6-1 = r3t5
rt
r
=
5
rt
1
1
a4b7
=
=
6-4b9-7
6
9
a
a²b²
ab
a5b3c8d4 = a5-2c8-4 = a3c4
2
7
4
9
7-3
9-4
4
5
abcd
b d
bd
Integer Exponents
Zero and Negative Exponents
Rule 5 a0 = 1
a≠0
x³
3-3
= x = xº
x³
*Summary any number (except for
0) or variable raised to the power
of zero = 1
Integer Exponents
Zero and Negative Exponents
-n = 1
x
Rule 6:
1 = xn
and
n
-n
1 x
1
X
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.
Integer Exponents
Zero and Negative Exponents
2 = 2a4
5a-4
5