Download presentation

Published byElvin Robertson Modified over 4 years ago

1
5.1 Linear Equations A linear equation in one variable can be written in the form: Ax + B = 0 Linear equations are solved by getting “x” by itself on one side of the equation Examples of non-linear equations:

2
5.1 Linear Equations Example: Solve by getting x by itself on one side of the equation. Subtract 7 from both sides: Divide both sides by 3:

3
5.1 Linear Equations Example: Renting a car for one day costs $20 plus $.25 per mile. How much would it cost to rent the car for one day if 68 miles are driven? $20 = fixed cost, $.25 68 = variable cost

4
5.1 Linear Equations A linear equation in two variables can be put in the form (called standard form): where A, B, and C are real numbers and A and B are not zero

5
5.1 Linear Equations Example (substitution): From the first equation we get y=2x-7, so substituting into the second equation:

6
**5.1 Linear Equations Example (elimination):**

Multiply the second equation by 3 to get: Adding equations you get:

7
**5.2 Graphs of Linear Functions**

x y 6 2 3 4 Graph by plotting points:

8
**5.2 Graphs of Linear Functions**

The slope of a line through points (x1,y1) and (x2,y2) is given by the formula:

9
**5.2 Graphs of Linear Functions**

A positive slope rises from left to right. A negative slope falls from left to right.

10
**5.2 Graphs of Linear Functions**

Finding the slope of a line from its equation Solve the equation for y. The slope is given by the coefficient of x Example: Find the slope of the equation.

11
**5.2 Graphs of Linear Functions**

Standard form: Slope-intercept form: (where m = slope and b = y-intercept)

12
**5.2 Graphs of Linear Functions**

Example: Put the equation 2x + 3y = 6 in slope-intercept form, determine the slope and intercept, then graph. Since b = 2, (0,2) is a point on the line. Since , go down 2 and across 3 to point (3,0) a second point on the line, then connect the two points to draw the line.

13
**5.2 Graphs of Linear Functions**

x y 2 3 Example: Graph the equation.

14
6.1 Special Products Special product: Example:

15
6.1 Special Products Difference of 2 squares: Example:

16
6.1 Special Products Squaring binomials: Examples:

17
6.1 Special Products Multiplying binomials using FOIL (First – Inner – Outer - Last): F – multiply the first 2 terms O – multiply the outer 2 terms I – multiply the inner 2 terms L – multiply the last 2 terms Combine like terms

18
6.1 Special Products Multiplying binomials: Example:

19
6.1 Special Products Multiplying binomials: Example:

20
6.1 Special Products Multiplying two polynomials (note: the book does this by grouping and using special products):

21
**6.2 Factoring: Common Factor and Difference of Squares**

Finding the Greatest Common Factor: Factor – write each number in factored form. List common factors Choose the smallest exponents – for variables and prime factors Multiply the primes and variables from step 3 Always factor out the GCF first when factoring an expression

22
**6.2 Factoring: Common Factor and Difference of Squares**

Example: factor 5x2y + 25xy2z

23
**6.2 Factoring: Common Factor and Difference of Squares**

Example: Note: the sum of 2 squares (x2 + y2) cannot be factored.

24
**6.2 Factoring: Common Factor and Difference of Squares**

Factor by Grouping – Introductory Example: Note: this will be covered in more detail in the next section.

25
**6.2 Factoring: Common Factor and Difference of Squares**

Factoring by grouping Group Terms – collect the terms in 2 groups that have a common factor Factor within groups Factor the entire polynomial – factor out a common binomial factor from step 2 If necessary rearrange terms – if step 3 didn’t work, repeat steps 2 & 3 until you get 2 binomial factors

26
**6.2 Factoring: Common Factor and Difference of Squares**

Example: This arrangement doesn’t work. Rearrange and try again

27
6.3 Factoring Trinomials Factoring x2 + bx + c (no “ax2” term yet) Find 2 integers: product is c and sum is b Sign hints: Both integers are “+” if b and c are “+” Both integers are “-” if c is “+” and b is “-” One integer is “+” and one is “-” if c is “-”

28
6.3 Factoring Trinomials Example:

29
6.3 Factoring Trinomials Factoring ax2 + bx + c by using FOIL (in reverse) The first terms must give a product of ax2 (pick two) The last terms must have a product of c (pick two) Check to see if the sum of the outer and inner products equals bx Repeat steps 1-3 until step 3 gives a sum = bx

30
6.3 Factoring Trinomials Example:

31
6.3 Factoring Trinomials Box Method (not in book):

32
6.3 Factoring Trinomials Box Method – keep guessing until cross-product terms add up to the middle value

33
6.3 Factoring Trinomials Perfect square trinomials: Examples:

34
**6.3 Factoring Trinomials Factoring ax2 + bx + c by grouping**

Multiply a times c Find a factorization of the number from step 1 that also adds up to b Split bx into these two factors multiplied by x Factor by grouping (always works)

35
6.3 Factoring Trinomials Example: Split up and factor by grouping

36
**6.4 Sum and Difference of Cubes**

Example:

37
**6.4 Sum and Difference of Cubes**

Sum of 2 cubes: Example:

38
**Summary of Factoring Summary of Factoring**

Factor out the greatest common factor Count the terms: 4 terms: try to factor by grouping 3 terms: check for perfect square trinomial. If not a perfect square, use general factoring methods 2 terms: check for difference of 2 squares, difference of 2 cubes, or sum of 2 cubes Can any factors be factored further?

39
6.5 Equivalent Fractions Polynomial Fraction– has the form: where P and Q are polynomials with Q not equal to zero.

40
6.5 Equivalent Fractions Lowest terms – A fraction P/Q is in lowest terms if the greatest common factor of the numerator and the denominator is 1. Fundamental property of fractions – If P/Q is a polynomial fraction and if K represents any polynomial where K 0, then:

41
**6.5 Equivalent Fractions Example: Write the fraction in lowest terms:**

Factor: By the fundamental property: The fraction is undefined for:

42
**6.6 Multiplication and Division of Fractions**

Multiplying Fractions– product of two fractions is given by: Dividing Fractions– quotient of two fractions is given by:

43
**6.6 Multiplication and Division of Fractions**

Multiplying or Dividing Fractions: Factor completely Multiply (multiply by reciprocal for division) Write in lowest terms using the fundamental property

44
**6.6 Multiplication and Division of Fractions**

Example - multiply: Factor: Cancel to get in lowest terms:

45
**6.6 Multiplication and Division of Fractions**

Example - divide: Factor: Cancel to get in lowest terms:

46
**6.7 Addition and Subtraction of Fractions**

Finding the least common denominator for rational expressions: Factor each denominator List the factors using the maximum number of times each one occurs Multiply the factors from step 2 to get the least common denominator

47
**6.7 Addition and Subtraction of Fractions**

Find the LCD for: Factor both denominators The LCD is the product of the largest power of each factor:

48
**6.7 Addition and Subtraction of Fractions**

Adding Fractions: If and are fractions, then Subtracting Fractions: If and are fractions, then

49
**6.7 Addition and Subtraction of Fractions**

Adding/Subtracting when the denominators are different fractions: Find the LCD Rewrite fractions – multiply top and bottom of each to get the LCD in the denominator Add the numerators (the LCD is the denominator Write in lowest terms

50
**6.4 Adding/Subtracting Rational Expressions**

Factor denominators to get the LCD: Multiply to get a common denominator: Add and simplify:

51
**6.7 Addition and Subtraction of Fractions**

Complex Fraction – a fraction with fractions in the numerator, denominator or both To simplify a complex fraction (method 1): Write both the numerator and denominator as a single fraction Change the complex fraction to a division problem Perform the division by multiplying by the reciprocal

52
**6.7 Addition and Subtraction of Fractions**

Example: Write top and bottom as a single fraction Change to division problem Multiply by the reciprocal and simplify

53
**6.7 Addition and Subtraction of Fractions**

To simplify a complex fraction (method 2): Find the LCD of all fractions within the complex fraction Multiply both the numerator and the denominator of the complex fraction by this LCD. Write your answer in lowest terms

54
**6.7 Addition and Subtraction of Fractions**

Example: Find the LCD: the denominators are 4, 8, and x so the LCD is 8x. Multiply top and bottom by this LCD. Simplify:

55
**6.8 Equations Involving Fractions**

Multiply both sides of the equation by the LCD Solve the resulting equation Check each solution you get – reject any answer that causes a denominator to equal zero.

56
**6.8 Equations Involving Fractions**

Solve: Factor to get LCD LCD = x(x - 1)(x + 1) Multiply both sides by LCD

57
**6.8 Equations Involving Fractions**

Example (continued): Solve the equation Check solution

58
**6.8 Equations Involving Fractions**

Distance, Rate, and time: Rate of Work - If one job can be completed in t units of time, then the rate of work is:

59
**6.8 Equations Involving Fractions**

Example: If the same number is added to the numerator and the denominator of the fraction 2/5, the result is 2/3. What is the number? Equation Multiply by LCD: 3(5+x) Subtract 2x and 6

60
**6.8 Equations Involving Fractions**

Example: It takes a mail carrier 6 hr to cover her route. It takes a substitute 8 hr. How long does it take if they work together? Table: Equation: Multiply by LCD: 24 Solve: Rate Time Part of Job Done Regular 1/6 x x/6 Substitute 1/8 x/8

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google