Presentation on theme: "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."— Presentation transcript:
1 5.1 Linear EquationsA linear equation in one variable can be written in the form: Ax + B = 0Linear equations are solved by getting “x” by itself on one side of the equationExamples of non-linear equations:
2 5.1 Linear EquationsExample: 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 EquationsExample: 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 EquationsA 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 EquationsExample (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 xy6234Graph 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 equationSolve the equation for y.The slope is given by the coefficient of xExample: 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 xy23Example: Graph the equation.
15 6.1 Special ProductsDifference of 2 squares:Example:
16 6.1 Special ProductsSquaring binomials:Examples:
17 6.1 Special ProductsMultiplying binomials using FOIL (First – Inner – Outer - Last):F – multiply the first 2 termsO – multiply the outer 2 termsI – multiply the inner 2 termsL – multiply the last 2 termsCombine like terms
18 6.1 Special ProductsMultiplying binomials:Example:
19 6.1 Special ProductsMultiplying binomials:Example:
20 6.1 Special ProductsMultiplying 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 factorsChoose the smallest exponents – for variables and prime factorsMultiply the primes and variables from step 3Always 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 groupingGroup Terms – collect the terms in 2 groups that have a common factorFactor within groupsFactor the entire polynomial – factor out a common binomial factor from step 2If 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 TrinomialsFactoring 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 “-”
29 6.3 Factoring TrinomialsFactoring 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 bxRepeat steps 1-3 until step 3 gives a sum = bx
34 6.3 Factoring Trinomials Factoring ax2 + bx + c by grouping Multiply a times cFind a factorization of the number from step 1 that also adds up to bSplit bx into these two factors multiplied by xFactor by grouping (always works)
35 6.3 Factoring TrinomialsExample:Split up and factor by grouping
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 factorCount the terms:4 terms: try to factor by grouping3 terms: check for perfect square trinomial. If not a perfect square, use general factoring methods2 terms: check for difference of 2 squares, difference of 2 cubes, or sum of 2 cubesCan any factors be factored further?
39 6.5 Equivalent FractionsPolynomial Fraction– has the form: where P and Q are polynomials with Q not equal to zero.
40 6.5 Equivalent FractionsLowest 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 completelyMultiply (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 denominatorList the factors using the maximum number of times each one occursMultiply 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 denominatorsThe 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, thenSubtracting Fractions: If and are fractions, then
49 6.7 Addition and Subtraction of Fractions Adding/Subtracting when the denominators are different fractions:Find the LCDRewrite fractions – multiply top and bottom of each to get the LCD in the denominatorAdd the numerators (the LCD is the denominatorWrite 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 bothTo simplify a complex fraction (method 1):Write both the numerator and denominator as a single fractionChange the complex fraction to a division problemPerform the division by multiplying by the reciprocal
52 6.7 Addition and Subtraction of Fractions Example:Write top and bottom as a single fractionChange to division problemMultiply 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 fractionMultiply 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 LCDSolve the resulting equationCheck 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 equationCheck 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?EquationMultiply 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: 24Solve:RateTimePart of Job DoneRegular1/6xx/6Substitute1/8x/8