# 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.

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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:

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:

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

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

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

5.1 Linear Equations Example (elimination):
Multiply the second equation by 3 to get: Adding equations you get:

5.2 Graphs of Linear Functions
x y 6 2 3 4 Graph by plotting points:

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

5.2 Graphs of Linear Functions
A positive slope rises from left to right. A negative slope falls from left to right.

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.

5.2 Graphs of Linear Functions
Standard form: Slope-intercept form: (where m = slope and b = y-intercept)

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.

5.2 Graphs of Linear Functions
x y 2 3 Example: Graph the equation.

6.1 Special Products Special product: Example:

6.1 Special Products Difference of 2 squares: Example:

6.1 Special Products Squaring binomials: Examples:

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

6.1 Special Products Multiplying binomials: Example:

6.1 Special Products Multiplying binomials: Example:

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

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

6.2 Factoring: Common Factor and Difference of Squares
Example: factor 5x2y + 25xy2z

6.2 Factoring: Common Factor and Difference of Squares
Example: Note: the sum of 2 squares (x2 + y2) cannot be factored.

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.

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

6.2 Factoring: Common Factor and Difference of Squares
Example: This arrangement doesn’t work. Rearrange and try again

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 “-”

6.3 Factoring Trinomials Example:

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

6.3 Factoring Trinomials Example:

6.3 Factoring Trinomials Box Method (not in book):

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

6.3 Factoring Trinomials Perfect square trinomials: Examples:

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)

6.3 Factoring Trinomials Example: Split up and factor by grouping

6.4 Sum and Difference of Cubes
Example:

6.4 Sum and Difference of Cubes
Sum of 2 cubes: Example:

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?

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

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:

6.5 Equivalent Fractions Example: Write the fraction in lowest terms:
Factor: By the fundamental property: The fraction is undefined for:

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:

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

6.6 Multiplication and Division of Fractions
Example - multiply: Factor: Cancel to get in lowest terms:

6.6 Multiplication and Division of Fractions
Example - divide: Factor: Cancel to get in lowest terms:

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

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:

6.7 Addition and Subtraction of Fractions
Adding Fractions: If and are fractions, then Subtracting Fractions: If and are fractions, then

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

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

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

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

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

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:

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.

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

6.8 Equations Involving Fractions
Example (continued): Solve the equation Check solution

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:

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

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

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