Chapter 3 Solving Equations

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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
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 5 4 - 3
1 1
3
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
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.
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Solving Equations 3y 27
Using the Multiplication Method we divide by the
coefficient, which is the same as multiplying by
?
3y 27
3 3
y 9
Check your solution
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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
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Solving Equations 2 Step 6x 12 36
6x 12 36 - 12 -12
Addition Method
6x 24
6x 24
Multiplication Method
6 6
x 4
Check
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Basic Percent EquationsPercent Base AmountP
B A
20 of what number is 30
multiply
equals
B
.2 B 30
B 150
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Basic Percent EquationsPercent Base AmountP
B A
What Percent of 80 is 70
P multiply equals
P 80 70
P .875
Convert to percentage.
P 87.5
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Basic Percent EquationsPercent Base AmountP
B A
25 of 60 is what?
multiply
equals
amount
.25 60 A
15 A
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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
• thats on the same side as the
• variable term to each side.

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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.
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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 Smallest variable term
-3x -3x 4x 8 -10
8 8 4x -2
Add the opposite of the constant
4 4
Divide by the Coefficient.
x -½
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Ex. 2 Solving Equations -24 (3b 2) 5
2(3b 6)
-24 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 and reduced
b ¼
CHECK
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Translating Sentences into Equations
Equation-equality of two mathematical
expressions.
Key words that mean
equals is is equal to amounts to represents
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Ex. Translate five less than a number is
thirteen
n - 5

13
n 18
Solve
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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
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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
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Addition and Subtraction
Polynomials can be added vertically or
horizontally.
Collect like terms
Horizontal Format
Ex. ( 3x³ - 7x 2) (7x² 2x 7)
3x³
7x²
- 5x
- 5
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Addition and Subtraction
Vertical Format
Ex. ( 3x³ - 7x 2) (7x² 2x 7)
³ ² ¹ º 3x³ - 7x 2
7x² 2x 7
Organized in columns by the degree
3x³7x² - 5x - 5
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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
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Subtraction
Vertical Format (-4w³ 8w 8) (3w³ - 4w² - 2w
1)
Change subtraction to the addition of the
opposite
³ ² ¹ º -4w³
8w - 8 -3w³ 4w² 2w 1
-7w³ - 4w² 6w - 9
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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 nm when
multiplying similar bases add
the powers.
Ex. y4 y y3 y 413 y8
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Multiplying Monomials
Ex. (8m³n)(-3n5)
Multiply the coefficients, Multiply similar
bases by adding the powers together
-24m3n6
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Simplify powers of Monomials
(x4)3 x4 x4 x4 x4 4 4 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 513x23y33 125x6y9
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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
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Multiplication of Polynomials
Distribute and follow Rule 1
-3a(4a² - 5a 6)
-12a³
15a²
- 18a
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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.
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Multiplication of two Polynomials
Ex. (y 2)(y² 3y 1)
Multiply y to every term. Multiply 2 to every
term.
y³
3y²
y
- 2y²
- 2
- 6y
Combine like terms.
y³ y² - 5y - 2
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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.
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Multiply two Binomials
Ex. (2x 3)(x 5)
O
F
F
O
I
L
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(2x 3)(x 5)
2x²
10x
3x
I
L
Collect Like Terms
2x² 13x 15
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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
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Sum and Difference of Binomials
(2x 3)(2x 3)
Square the term 2x
-
4x²
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Square the term 3
Minus sign between the terms
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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
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Square of a Binomial
Ex. (5x 3)²
25x²
30x
9
Square 5x
Multiply 5x and 3 then times by 2
Square the 3
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Square of a Binomial
(4y 7)²
16y²
- 56y
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Square the 4y
Mutiply the 4y and 7 then times by 2
Square the 7
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Integer ExponentsDivide Monomials
x5 x2
xxxxx xx
x³


Rule 4 xm xn
When m gt n
Xm-n

xm xn
1 xn-m
When n gt m

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Integer ExponentsDivide 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


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Integer ExponentsZero and Negative Exponents
Rule 5 a0 1 a ? 0
x³ x³
x3-3
xº
Summary any number (except for 0) or variable
raised to the power of zero 1
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Integer ExponentsZero 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 its
negative exponent and move them from the
numerator to the denominator and the sign of the
exponent changes.
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Integer ExponentsZero and Negative Exponents
2 5a-4
2a4 5