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