chapter 3 solving equations introduction to equations equation: equality of two mathematical...

Chapter 3 Solving EquationsIntroduction to Equations

Equation: equality of two mathematical expressions. =

9 + 3 = 123x – 2 = 10y² + 4 = 2y - 1

Solution to an equation, is the numberwhen 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 - 31 = 1

Solve an equation Addition Propertyr – 6 = 14r – 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 equations + 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 Equations3y = 27

3y = 27 3 3

Using the MultiplicationMethod we divide by the coefficient, which is the same as multiplying by ⅓

y = 9

Check your solution

Solving Equations

85

4x

85

4x

4

5

4

5

Using the multiplicationmethod we multiply thereciprocal of the coefficient to both sides.

X = 10 Check 4/5(10) = 8 8 = 8

Solving Equations: 2 Step6x + 12 = 36 6x + 12 = 36 - 12 -12 6x = 24

6x = 24 6 6

x = 4

Addition Method

MultiplicationMethod

Check

Basic Percent EquationsPercent • Base = Amount

P • B = A20% of what number is 30

multiply equals

.2 • B = 30

B = 150

B

Basic Percent EquationsPercent • Base = Amount

P • B = A

What Percent of 80 is 70

P multiply equals

P • 80 = 70P = .875P = 87.5% Convert to percentage.

Basic Percent EquationsPercent • 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 symbols2. 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 Equations3x – 4(2 – x) = 3(x – 2) - 43x – 8 + 4x = 3x – 6 – 4 Distribute7x – 8 = 3x - 10 Collect like terms

-3x -3x 4x – 8 = -10

Add opposite of theSmallest variable term

+ 8 + 8 4x = -2

Add the opposite of the constant

4 4 x = -½ Divide by the

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 - 126b – 4 = -6b - 7

12b – 4 = -7 12b = -3

b = ¼

CollectedAdded 6b

Added 4Divided by 12 and reducedCHECK

Translating Sentences into Equations

Equation-equality of two mathematical expressions.

Key words that mean =equalsisis equal toamounts to represents

Ex. Translate: “five less than a number is thirteen”

n - 5 = 13

Solve n = 18

Translate Consecutive IntegersConsecutive integers are integers thatfollow one another in order.

Consecutive odd integers- 5,7,9

Consecutive even integers- 8,10,12

CHAPTER 4 POLYNOMIALSPolynomial: a variable expressionin which the terms are monomials.

Monomial: one term polynomial

Binomial: two term polynomial

Trinomial: Three term polynomial

5, 5x², ¾x, 6x²y³ Not: or 3r

xy x

5x² + 7

3x² - 5x + 8

Addition and Subtraction

Polynomials can be added vertically or horizontally.

Ex. ( 3x³ - 7x + 2) + (7x² + 2x – 7)Horizontal Format Collect like terms

3x³ + 7x² - 5x - 5

Addition and Subtraction

Vertical FormatEx. ( 3x³ - 7x + 2) + (7x² + 2x – 7) ³ ² ¹ º 3x³ - 7x + 2 +7x² + 2x – 7

Organized incolumns by thedegree

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)

³ ² ¹ º -4w³ + 8w - 8-3w³ + 4w² +2w + 1

Changesubtraction tothe addition ofthe 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

RULE 1 xn • xm = x n+m

when multiplying similar bases add the powers. Ex. y4 • y • y3 = y 4+1+3 = y8

Multiplying Monomials

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

-24m3n6

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 = 51•3x2•3y3•3 = 125x6y9

Simplify Monomials Continue

Ex. (ab²)(-2a²b)³

(ab²)(-8a6b³)

Rule 3: Multiplythe outside powerto inside powers.

-8a7b5

Rule 1: multiply theMonomials by adding

the exponents

Multiplication of Polynomials

-3a(4a² - 5a + 6)Distribute and follow

Rule 1

-12a³ + 15a² - 18a

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)

y³ + 3y² + y

- 2y² - 6y - 2

y³ + y² - 5y - 2

Multiply y toevery term.Multiply –2 toevery term.

Combine like terms.

Multiply two Binomials

The product of two binomials can befound 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)

(2x + 3)(x + 5) =

F

2x² F O

+10xO

I

+3x I

L

+15 L

Collect Like Terms

2x² + 13x + 15

Special Products of Binomials

Sum and Difference of two Binomials

(a + b)(a – b)

a² b² -

Square the first term

Square the secondterm

Minus sign between the products

Sum and Difference of Binomials

(2x + 3)(2x – 3)

4x² 9-Square the term 2x

Square the term 3

Minus sign between the terms

Square of a Binomial

(a + b)² = (a + b)(a + b) Then FOILOr use the short cut

(a + b)² = a²

1. Square 1st term ab times 2 =

+ 2ab

2. Multiply terms and times by 2.

+ b²

3. Square 2nd term

Square of a Binomial

Ex. (5x + 3)² =

Square 5x

25x²

Multiply 5x and 3then times by 2

+ 30x

Square the 3

+ 9

Square of a Binomial

(4y – 7)² = 16y² - 56y + 49

Integer ExponentsDivide Monomials

x5

x2 = x•x•x•x•x

x•x = x³

Rule 4 xm

xn = Xm-n When m > n

xm

xn= 1

xn-m

When n > m

Integer ExponentsDivide Monomials

r8t6

r5t= r8-5t6-1 = r3t5

a4b7

a6b9=

1 a6-4b9-7

= 1 a²b²

a5b3c8d4

a2b7c4d9= a5-2c8-4

b7-3d9-4= a3c4

b4d5

Integer ExponentsZero and Negative Exponents

Rule 5 a0 = 1 a ≠ 0

x³x³

= x3-3 = xº

*Summary any number (except for 0) or variable raised to the power of zero = 1

1

Integer ExponentsZero and Negative Exponents

Rule 6: x-n = 1 xn and 1

X-n

= xn

If we make everything a fraction,1

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 ExponentsZero and Negative Exponents

2 5a-4

= 2a4

5