Exponential Properties 0.1 Topic: Exponential Properties

Transcription

1 Ns Exponential Properties 0.1 Topic: Exponential Properties Date: Objectives: SWBAT (Simplify and Evaluate Expressions using the Exponential LAWS) Main Ideas: Assignment: LAW Algebraic Meaning Example Product of Powers x a x b = x a+b x 2 y 4 x 5 y = Quotient of Powers {x 0} x a = xa b xb 4x 7 y 8 2x 5 y = Exponential LAWS Negative Exponents {x 0} x a = 1 x a 1 and = xa x a b 5 = 1 h 9 = Power of a Power (x a ) b = x ab (v 6 ) 7 = Power of a Product (xy) a = x a y a (x 5 y 7 ) = Power of a Quotient {x 0, y 0} Zero Power {x 0} ( x y ) a = xa y a ( x a y ) = ( y a x ) x 0 = 1 = ya x a ( x5 y 7 xy 5 ) = ( (x4 y 5 ) 4 0 (xy 5 ) ) =

2 Simplify using no negative exponents. (2x y )( 7x 5 y 6 ) 15c 5 d c 2 d 7 Simplifying Monomials ( a 4 ) ( 2x y 2 ) 5 Degree and Naming Polynomials Definition of a polynomial is a function in the form f(x) = a n x n + a n 1 x n a 1 x + a 0 a n 0, All exponents are whole # s and all coefficients are real # s a n is the Leading Coefficient (or LC) n is the Degree of the polynomial (highest degree of any term in polynomial) a 0 is the constant term 1 4 x4 y 8x 5 x + x + 4 x + 2x ( x 2 x + 4) (x 2 + 2x + 5) 4 x2 (6x 2 + 9x 12) Other Examples (x 2 + 4x + 16)(x 4) ab(4a 5b) + 4b 2 (2a 2 + 1)

3 Exponential Properties 0.1 Simplify. Assume that no variable equals 0. (5x y 5 )(4xy ) ( 2b c)(4b 2 c 2 ) y z 5 y 2 z 7x 5 y 5 z 4 21x 7 y 5 z 2 Depth Of Knowledge 2xy x 4 y 2 ( x 2 4x 4 y 6) ( 5a 7 b 2 ab 6 ) Determine whether each expression is a polynomial. If yes, identify the degree. 2x 2 x + 5 5mp n 2 2g h a 11 m 7 Simplify. (6a 2 + 5a + 10) (4a 2 + 6a + 12) 4x(2x 2 + y) (a + b)(a ab b 2 ) (a + b)(2a + b)(2x y)

4 Ns Rational Exponents 0.2 Topic: Rational Exponents Date: Objectives: SWBAT (Simplify and Evaluate Expressions with Rational Exponents) Main Ideas: Assignment: Rational Exponents b 1 n Definition of b 1 n: Words: A number raised to the 1 n power is equal to the n th root of that number or base. Examples: Write in radical notation. Examples: 1 2 = = = 2 1) ) ( ) 1 ) Write with rational exponents. 5 4) 7 Simplify. 5) 10 6) 2 Algebra: If b > 1, and n is an integer, where n 2, then b 1 n n = b. 9 7) 4 1 8) Rational Exponents b m n Definition of b m n : Words: A number raised to the m n power is equal to the n th root of that number or base raised to the m th power. Examples: 8 2 = ( 8) 2 or 8 2 = 8 2 Algebra: If b > 1, and m and n are integer, where m 1 and n 2, then b m n n = b m n = ( b) m. Examples: 9) Write ( ) 2 5 in radical notation 10) Write x in rational notation Simplify. 11) (81) ) (16) 4 1) ( 64) 2

5 Simplifying Radicals To simplify the square root of a number, create a factor tree and any two identical numbers may come out of the radical. Buddy system or find the largest perfect square factor of that number. If it is not perfect or part of a buddy then it stay under the radical. Buddy System Largest Perfect Square Simplifying Cube Roots To simplify a cube root use the same process as a square root, except of the same numbers are needed to come out or find a perfect cube. Buddy System Largest Perfect Cube The same is true for any other power outside the radical. Any Root

6 Variables in Radicals Rational Exponents 0.2 If the radical contains variables: Divide power of variable by the root. The quotient is the new power of the variable outside the radical If there is a remainder, that is the power of the variable still under the radical x 10 y x 8 y 9 (4x + 7) 24 y 9 (x 8) 8 EXAMPLES: Write all radicals as exponentials, and then apply the rules for rational exponents. Leave answers in exponential form. x x x 2 4 EXAMPLES: Simplify Write your answer in simplest root and exponential form. (x y 7 z 8 ) 1 2 (8x 5 ) 1 (2x 5 ) ( 125x 4 x) 2 Examples EXAMPLES: Evaluate each exponential ( 64) 2 ( ) 2

7 Ns Radical Operations Dude! 0. Topic: Operations with Radicals Date: Objectives: SWBAT (Perform Basic Operations with Radicals and Simplify Answer) Main Ideas: Rationalize Denominator Assignment: When a radical contains an expression that is not a perfect root, for example, the square root of or cube root of 5, it is called an irrational number. So, in order to rationalize the denominator, we need to get rid of all radicals that are in the denominator. Example: a x Product Property n For any real numbers a and b and any integer n > 1, ab and b are nonnegative or if n is odd. Examples: 2 8 = 16 or 4 9 = 27 or 20 = 4 5 = 2 5 or 2 5 n = a n b, if n is even and a Examples Simplify Simplify 25a 4 b Simplify Simplify 125m 0 p 20 2 ( )

8 Quotient Property For any real number a and b 0 and any integer n > 1, a b defined. Examples: 25 4 = 25 4 or 5 2 x6 8 = x6 or x2 8 2 n n = a n b, if all roots are 27 = 27 = 9 or Simplify Simplify Examples y8 x 7 Simplify 8x 6 2 9x Simplify 54 6 Simplifying Radicals A radical expression is in simplest form when the following conditions are met: The index n is as small as possible and all like radicals are combined The radicand contains no factors (other than 1) that are n th powers of an integer or polynomial The radicand contains no fractions No radicals appear in the denominator Index n a Radical Radicand Simple Examples

9 Examples Different Operations Radical Operations Dude! 0. Simplify Simplify Simplify 5 100a 2 10a Simplify Simplify Simplify 16a a ( )(5 7) Conjugates Conjugates are used to simplify an expression with a radical sign in them to form a difference of squares. (form: a b + c d and a b c d) ( 7)( + 7) Simplify Examples Upper

10 Ns Rational Expo in Expressions 0.4 Topic: Rational Exponents in Expressions Date: Objectives: SWBAT (Simplify Expression with Rational Exponents) Main Ideas: Assignment: Rational Exponents For any real number b and any positive integer n, b 1 n n = b 1, except when b < 0 and n is even. When b < 0 and n is even, then a complex root may exist. Examples: Deep Look into = 27 or Radical Form/Exponential Form Write in Radical Form a 1 7 Write in Radical Form x 1 Write in Exponential Form w Write in Exponential Form 5 z Evaluating 2 Methods

11 Evaluate Your Turn Any Method Any Method 16 4 Rational Exponents Part 2 For any real nonzero number b, and any integers x and y, with y > 1, x y by = b x y = ( b) x, except when b < 0 and y is even. When b < 0 and y is even, a complex root may exist. Examples: Two Ways to Look at this: 27 2 = ( 27) 2 = () 2 or Simplify Simplify Your Turn (25) 2

12 Rational Expo in Expressions 0.4 Fully Simplified An expression with rational exponents is fully simplified when all of the following conditions are met. It has no negative exponents in the numerator It has no exponents that are negative integers in the denominator It is not a complex fraction The index of any remaining radical is the least number possible Simplify y 1 7 y 4 7 Simplify (x 1 y 1 2) 6 Simplifying Rational Expressions Simplify x 1 5 x 2 5 Simplify y 4 Simplifying Radical Expressions Simplify Simplify 6 4x 4

13 Simplify y y Simplify 4 2 Simplify 6 16x 2 Simplify x x

14 Ns Imaginary Numbers 0.5 and 0.6 Topic: Operations with Radicals Date: Objectives: SWBAT (Perform Basic Operations with Imaginary Numbers) Main Ideas: Explore Imaginary Numbers Assignment: Two Different Answers Reminders (?? ) a b = a b or ab or ab = a b a b = a b or a b = a b First Must Accept that there are values that are Imaginary To be able to solve or simplify questions like the ones below, we must have something that was squared to get a negative value: x = 0 (x + ) 2 = 1 5 ± 25 (imaginary #) 2 = negative value An imaginary numbers is one that when squared gives a negative value. So far we have excluded expressions such as 16 because it is not a real number; there is no real number whose square is -16. This number is part of the complex number system. The complex number system allows us to solve equations such as x = 0. Imaginary Unit: The imaginary unit, written as i, is the number whose square is 1. That is, i 2 = 1 and i = 1 Example: Write square root of negative number in terms of i. 16 = 1 16 = 1 16 = i 4, or 4i (i notation) 4i is the product of a real number and i (pure imaginary)? s #-line Write using i notation Deeper look into the rotation around the real number line. Imaginary axis Real Number Line

15 i sequence i = i 2 = i = i 4 = i 5 = i 6 = i 7 = i 8 = i 9 = i 10 = i 11 = i 12 = Did you find a pattern? If so, write a summary for what you discovered. i 45 = i 81 =? s i 202 = i 1012 = Complex Numbers Big Picture Real Numbers Rational Irrational Integers Whole Numbers Natural Numbers Imaginary Numbers Def n of a If a is a positive real number, then the principal square root of negative a is the imaginary number i a. a = i a Pure Imaginary Number of the form 6i, 2i, and i are called pure imaginary numbers. Pure Imaginary Numbers are square roots of negative numbers. For any positive real number b, b 2 = b 2 1 or bi

16 Imaginary Numbers 0.5 and 0.6 Simple Operations with i 2i + i + i i 2 i i 9i 6i 7i i ( 2i) Simplifying using Def n of i ( 16) ( 125) i i 5i ( 6i) Other Operations i 7 i 5 2i (2i) 2 ( i 2 ) 1 4i( 6i) 2 5 i 18 2i

17 x 2 = 100 x 2 9 = 0 Do you know how to solve this? x 2 = 25 x = 0

18 Ns ber Complex Numbers 0.7 Topic: Complex Numbers Date: Objectives: SWBAT (Perform Basic Operations with Complex Numbers) Main Ideas: Assignment: Complex Numbers A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit. It is a sum of an imaginary part (bi) and a real number(a). Example: i (standard form) Real numbers are a subset of complex numbers and can be written in the form of a complex number. Example: b = 0 16 = i Imaginary numbers are a subset of complex numbers and complex numbers are pure imaginary numbers if a = 0 and b 0. Example: i = 0 + i (pure imaginary) Examples Write in the form a + bi (Simplify first where necessary). Identify a and b. Then classify according to the most restrictive classification: complex, imaginary, rational, irrational, or integer ( i) 1 5 i 25 Sum and Difference of Complex Numbers Sum: Difference: (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) (c + di) = a + bi c di = (a c) + (b d)i Add or subtract as indicated. Write your answer in a + bi form. ( 5i) + (2 + 4i) (8 i) (2 i) (8 i) (2 i) 2 + 5i (4 + 7i)

19 Find the Product: Find the Product: Use same concept 6x ( x) 7i i Multiply Complex Numbers Patterns 7(5 2n) (x )(x + 2) Difference of Squares (2 + )(2 ) (4 i)(4 + i) Multiply. Write your answers in a + bi form. i(2 i) (2 5i)(4 + i) Square Double Square (Perfect ) (2 + ) 2 (4 i) 2 6i 8i i 5i 2i(4 9i) (2 + i)(1 + 4i) ( 2 2i)( 2 + 2i) ( 2i) 2 x = 0 x 2 4x + 7 = 0 Solve

20 Critical Thinking Complex Numbers 0.7 How are the following problems different? (2 + x)( 2x) (2 + i)( 2i) How are the following problems different? 2 + x ( 2x) 2 + i ( 2i)

21 Ns Complex Conjugates 0.8 Topic: Complex Conjugates Date: Objectives: SWBAT (Understand the basic concepts of Complex Numbers) Main Ideas: Review Examples Assignment: -Often in mathematics, it is helpful to write a radical expression such as without a radical 2 in the denominator. -The process of writing this expression as an equivalent expression but without a radical in the denominator is called rationalizing the denominator. Easy Math: 2 = = 8 12 Algebra: 2 = = = 6 4 = Rationalizing with a Sum or Difference of two terms in denominator -To rationalize a denominator that is a sum or difference of two terms, we use conjugates. -Conjugates create a difference of squares and eliminate any radicals or imaginary numbers by creating perfect squares Conjugates In Algebra, the conjugate of a binomial is another binomial formed by negating the second term of original binomial. The conjugate of x + y is x y, where x and y are real numbers. If y is imaginary, then process is termed complex conjugation: the complex conjugate of a + bi is a bi, where a and b are real. (a + bi) and (a bi) are complex conjugates of each other (a + bi)(a bi) = a 2 abi + abi (bi) 2 = a 2 + b 2 (sum of squares) Example: multiply (2 i) and its conjugate (2 + i) and (2 + 5) and its conjugate (2 5) (2 i)(2 + i) (2 + 5)(2 5) What do you notice about the product?

22 i 1 i 6 Examples 1 5i 1 5 2i i Find the reciprocal of i i i 5 1 2i i + 2 i 2 + i i Higher Level

23 Ns Equations and Inequalities 0.9 Topic: Equations and Inequalities Date: Objectives: SWBAT (Solve multi-step equations and inequalities) Main Ideas: Assignment: #1) Do you have any grouping symbols like ( ) or [ ]? Yes No Steps to Solving Linear #2) Do you have like terms on either side of the equal sign? #) Do you have variables on both sides of the equal sign? #4) Do you have constant on the side of the equation that the variable is on? #5) Do you have a coefficient attached to the variable other than 1? Example: Check: 2(4x 7) + 15 = 8 2(4( ) 7) + 15 = 8 Yes Yes Yes Yes No No No No Addition Property of Equality Subtraction Property of Equality Properties Multiplication Property of Equality Division Property of Equality Distributive Property Identity Infinitely Many Solutions No Solution x + 2 = x + 2 x = x + 1

24 x 5 = x + 2 7x Justification (x 2) + = 2(6 x) Justification Justify 5(2x 7) + 2 = (4 x) 12 Justification Fraction Bust Inequalities x = x What is the flip-flop rule of solving linear inequalities Justification

25 Equations and Inequalities 0.9 5(x 2) 9x (2x 4) 4(2x 1) > x 2(x 5) Compound Inequalities Examples Compound Inequalities are formed by joining two inequalities with a connective word such as and or or. Example: 2x < 4 and x 2 > 8 2x + > 5 or x + 2 < 5 The solution set of a compound inequality with the connective word and is the set of all elements that the inequalities have in common.this is called an intersection Solve: 2x < 6 and x + 2 > 4 The solution set of a compound inequality with the connective word or is the set of all elements for both inequalities.this is called a union 2x + > 7 or 4x 1 <

26 1 < x 5 < x > and 7 x < 4 Examples 4x > 7 or 4x + 5 < x 7 2

27 Ns Topic: Solving Quadratics Solving Quadratics 0.10 Date: Objectives: SWBAT (Solve Quadratics using ZPP and Quadratic Formula) Main Ideas: Assignment: Factoring Quadratics ZPP See Flow Map for Factoring Quadratics Zero Product Property: If the product of two factors is zero, then at least one of the factors must be zero. If ab = 0, then a = 0 or b = 0 (x 4)(x + 2) = 0 2x 2 x = 1 Examples 4x x = x x 2 4x + 4 = 0 (x 2)(x + 5) = 8 x + 4x 2 9x 6 = 0

28 No Real Solutions/Double Root/Two Solutions x 2 9 = 0 x = 0 x 2 20x = 0 x = 0 (x + 1) 2 = 0 x = 0 Quadratic Formula The solutions to a quadratic equation ax 2 + bx + c = 0, where a 0, can be found by using the Quadratic Formula (when in Standard Form): x = b ± b2 4ac 2a Solve x 2 2x = 5 by using the Quadratic Formula. Factoring vs. Formula Factoring x 2 2x = 5 Quadratic Formula x 2 2x = 5 Solve x 2 + x 0 = 0. Round to the nearest tenth if necessary. Factoring x 2 + x 0 = 0 Quadratic Formula x 2 + x 0 = 0

29 Solving Quadratics 0.10 Solve 2x 2 2x 5 = 0 by using the Quadratic Formula. Round to the nearest tenth if necessary. Rounded Answer 2x 2 2x 5 = 0 Exact Answer Form 2x 2 2x 5 = 0 Rounded vs. Exact Solve 5x 2 8x = 4 by using the Quadratic Formula. Round to the nearest tenth if necessary. Rounded Answer 5x 2 8x = 4 Exact Answer Form 5x 2 8x = 4 x 27 = 0 x = 0 Factoring and Solving Diff/Sum of Cubes 8x 1 = 0 64x + 27 = 0

30 Factor (a b) b Factor a + (a + b) Different But Same Factor x 6n + y n Factor x n + y n x 4 9x = 0 x 8 +4x 4 5 = 0 Using Quadratic Form u = something p p = 0 x 2 x 1 6 = 0

31 Ns Completing the Square 0.11/0.12 Topic: Completing the Square Date: Objectives: SWBAT (Solve Quadratics using the Complete the Square Method) Main Ideas: Completing the Square Assignment: Completing the Square Method is where you take: Take something like this And turn it into This ax 2 + bx + c = 0 a(x c) 2 + d = 0 Standard Form Vertex Form Steps: Example 1. Moves constant over 2x 2 + 8x + 10 = 0 2x 2 + 8x = Divide everything by the a (LC) 2x x 2 = Cut the middle term s coefficient in half and square your result 4. Add this value to both sides of the equation 5. Factor the Perfect Square Trinomial it will be (x + b 2 )2 6. Solve the remaining quadratic using the Square Root Property x 2 + 4x = 5 Algebra Actual ( b 2 2 ) ( ) = 4 x 2 + 4x + 4 = x 2 + 4x + 4 = 1 (x + 2)2 = 1 (x + 2) 2 = 1 x 2 2x 15 = 0 x 2 2x 1 = 2 Me, We, Two, You x 2 2x 2 = 4 x 2 + 4x =

32 Factoring vs. Completing the Square vs. Quadratic Formula Discuss with your neighbor the advantages and disadvantages of using one method over the other. You have equations below you can only use each method once.then explain below your answer why you chose your method. 2x 2 16x 12 = 0 x 2 + 4x 7 = 0 4x 2 7x 1 = 0 How about these? Any method x 4 x = x = 1 x 2 2 x + 5 x x 5 = 1

33 Ns Absolute Value Equations/Ineq 0.1 Topic: Absolute Value Equations/Inequalities Date: Objectives: SWBAT (Solve Absolute Value Equations and Inequalities) Main Ideas: Assignment: Evaluate a if a = 5 Evaluate Absolute Value Evaluate 17 b + 2 if b = 6 Evaluate 4a b + 2c if a = 2, b =, and c = 2 When solving absolute value equations, there are two cases to consider. Solving Absolute Value Equations Case 1 The expression inside the absolute value symbol is positive or zero Case 2 The expression inside the absolute value symbol is negative For any real numbers a and b, if a = b and b 0, then a = b of a = b. Example: d = 10 means that d (the expression inside) can have a value of 10 or 10 Steps to Solve: #1) Isolate the absolute value #2) Create one or two equations 1 if equal to 0, 2 if equal to a positive #) Set the expression inside the absolute value equal to the value or values.drop the absolute value bars #4) Solve equation or equations #5) Put solution or solutions in solution set

34 Absolute Value Equations/Ineq 0.1 Solve 2x 1 = 7 Solve p + 6 = 5 Me, We, Two, You Solve 5 8 2n = 75 Solve 6 1 5x 9 = 57 x + 10 = 4x 8 2x + 2 2x = x + Upper Level Constraints Extraneous Solutions

35 Absolute Value Equations/Ineq 0.1 Solving Absolute Inequalities Steps: 1. Get rid of inequality symbol (replace with = sign) and solve like an equation. 2. Your two solutions will break your number line into intervals.. Check values in each interval. 4. Your solution will either be an intersection or union. + n 2 > v r 5 7 Your Turn

36 Solving Exponentials 0.14 Topic: Solving Exponential Equations/Inequalities Date: Objectives: SWBAT (Solve Exponentials using the Change /Same Base Method) Main Ideas: Assignment: Property of Equality for Exponential Functions (or Same Base Method) Property of Equality for Exponential Functions In Words: Let b > 0 and b 1. Then b x = b y, if and only if x = y. Example: If x = 5, then x = 5. If x = 5, then x = 5. Solve for x. 4 2 = 4 x 15 x = x = = 10 x 6 8 = 6 2x 20 = x+ 100 = 100 x = 2 2x = 18 Change/Same Base Method Solve: Solve: x = = 4 x Solve: Solve: 8 = 2 x 9 2x 1 = 6x

37 Solve: 1 = 42x 1 25x Solve: 2x = 9 5x 4 Solve: 4 2n 1 = 1 64 Solve: 5 5x = 125 x+2 Solve: 2x+1 2x = x Solve: 4 2x+ = 1 Upper Level Solve: 1 2 x 2 = 2 Solve: 2 x = 2 2 Solve: x 5 = 27 9

38 Property of Inequality for Exponential Functions Solving Exponentials 0.14 Property of Inequality for Exponential Functions In Words: Let b x > b y, if and only if x > y, and b x < b y if and only if x < y. Example: If 2 x > 2 6, then x > 6. If x > 6, then 2 x > 2 6. Solve: 5 2x > x > 1 24

39 Ns Solving Radical Equations/Inequalities 0.15 Topic: Solving Radical Equations and Inequalities Date: Objectives: SWBAT (Solve Radical Equations and Inequalities) Main Ideas: Assignment: Write in Radical Form Evaluate Review Evaluate Simplify w 9 7 w 5 7 Steps to Solving Radicals Solving Radical Equations: Step #1) If the index of the root is even, identify the values of the variable for which the radicand is nonnegative (restricting domain or constraints). {x } Step #2) Isolate the radical sign on one side of the equation. x + 5 = x = 10 Step #) Raise each side of the equation to a power equal to the index of the radical to eliminate the radical. ( x ) 2 = (10) 2 x = 100 Step #4) Solve the resulting polynomial equation. Check your results (extraneous). x = = + x = 10 Check (10) + 5 = = = = 15

40 Solve y 2 1 = 5 Solve x 12 = 2 x Me, We, Two, You Solve x 2 + x 2 x = 1 Solve x + 5 = 1 x Solve y = 0 Solve 2y + 1 = 0 How do you like these two? Other Examples Solve 6 7( 5m + 4) 4 = 10 Solve 9( h + ) 4 = 77

41 Solving Radical Equations/Inequalities 0.15 Solving Radical Inequalities Solving Radical Inequalities: Step #1) If the index of the root is even, identify the values of the variable for which the radicand is nonnegative (restricting domain or constraints). + 5x 10 8 So, since 5x 10 0.this means that x 2 Step #2) Solve the inequality algebraically. + 5x 10 8 ( 5x 10) 2 (5) 2 5x x x 7 Step #) Test values (create three intervals using solution s value and the value that restricts the x-values/ domain ) to check your solution. Check: x = 0 x = 4 x = 9 Solve x Solve 1 + 7x > Your Turn

42 Ns Solving Radical Equations 0.16 Topic: Solving Radical Equations Date: Objectives: SWBAT (Solve Radical Equations using Rational Exponents) Main Ideas: Assignment: Review Solving Radicals with Rational Exponents What does 8 2 really mean? Steps: 1. Restrict the Domain if necessary (even power root) 2. Isolate the base with the rational exponent. Solve the remaining equation x 2 = 4 x = 10 x = 65 x = 92 (x ) 2 9 = 0 (x ) = 0 Upper Level (5 4x) = 0 (2x 5) 4 6 = 18

43 Solving Rational Equations 0.17 Topic: Solving Rational Equations Date: Objectives: SWBAT (Solve Rational Equations) Main Ideas: Assignment: Back to the Future Solve. No short-cuts x = x Now Fraction Bust and Solve. x = x Steps to Solve: #1) Restrict the Domain. #2) Find the LCD. #) Multiply whole equation by LCD to fraction bust. #4) Solve the resulting equation. #5) Check solutions for extraneous. Solve x 1 = 5x x Solve. x = 6 x Solve. 4 2y + 1 = y y y Start Easy

44 Solving Rational Equations Check for Understanding Solve. Solve. p 2 p 5 p + 1 x x 2 = = p2 7 p 1 + p 2 x + 4 2x x 2 + 2x 8

45 Solving Rational Equations 0.17 Solve x = p = 1 p 5 p + 4 p 2 5p More Examples Solve. 5x 20 x 2 9x x 6 = x 4 x 2 9x + 18

46 Solve. 5 n + 5n 2 = 4 n n 2 Last One I Promise (Extraneous Solutions)

47 SYSTEMS All Types 0.18 Topic: Solving Systems of Equations Date: Objectives: SWBAT (Solve Systems of Equations) Main Ideas: Assignment: What is the Solution to a system of equations? What is a system of equations? Types of Solutions: How many different ways can you get lines to cross? Types of Solutions One Solution Independent Consistent No Solution Inconsistent Infinitely Many Solutions Dependent Consistent Solve each system of equations by GRAPHING. Clearly Identify your solution. 1. x + y = x + 2y = 14 x + 2y = 6 2. x = 4 pg. 177

48 SYSTEMS All Types 0.18 Solve each system of equations by SUBSTITUTION. Clearly Identify your solution.. y = 7x + 6 4x y = x 7y = 21 2x 14y = x 5y = 15 x 4y = y = 5 8x + 5y = 17 Solve each system of equations by ELIMINATION. Clearly Identify your solution. 7. x y = 10 x 6y = 25 2x + 2y = x 2y = x + 6y = 27 x + 2y = x + 5y = 22 7x y = 2 pg. 178

49 SYSTEMS All Types 0.18 A system of linear equations in two variables has a solutions as an order pair (x, y), a solution of an equation in three variables is an order triple (x, y, z). Example: x 2y + z = 6 { x + y 2z = 2 2x y + 5z = 1 2x y z = 1 { x + 4y + z = 2 4x 6y 2z = 5 Your Turn pg. 179

50 Systems of Linear Inequalities SYSTEMS All Types 0.18 A system of linear inequalities is two or more linear inequalities working together on one coordinate plane. The solution to a system of linear inequalities is the of points that satisfies both or all the inequalities. x + y > x y > 5 x + y < y x x < 4 x + 2y y < 5x + 6 y 2x 1 pg. 180

51 SYSTEMS All Types 0.18 Linear and Quadratic Systems Upper Level Which one doesn t belong? y = x 2 5x + 7 y = 2x + 1 y x 2 = 7 5x 4y 8x = 21 Solve y = x y = x + 1 y = 5x 20 y = x 2 5x + 5 pg. 181

52 Ns ber Operations with Matrices 0.19 Topic: Operations with Matrices Date: Objectives: SWBAT (Perform Basic Operations with Matrices) Main Ideas: Matrices Assignment: A matrix is a collection of numbers ordered by rows and columns. For example, the following is a matrix: [ ] This matrix has 2 rows and columns and is referred to as a 2x matrix. The elements in the matrix are identified as the following: [ a 11 a 12 a 1 a 21 a 22 a 2 ] The element a 12 means element locate in row 1 column 2. Adding and Subtracting To add or subtract matrices you must have the same dimensions.you add or subtract the corresponding elements. A + B = A + B [ a 11 a 12 a 21 a ] + [ b 11 b 12 ] [ a 11 + b 11 a 12 + b 12 ] Square Matrix 22 b 21 b 22 a 21 + b 21 a 22 + b 22 A + B = A B [ a 11 a 12 a 21 a ] + [ b 11 b 12 ] [ a 11 b 11 a 12 b 12 ] Square Matrix 22 b 21 b 22 a 21 b 21 a 22 b 22 Example [ ] + [ ] = [ ] = [ 1 + ( 9) ] [ ] [ 4 ] [ 9 ] + [ ] 8 [ ] [ ] Examples

53 You can multiply matrices by a constant called a scalar by multiplying all elements in matrix by that constant. Scalar Multiplication Symbols k A = ka k [ a b kb ] = [ka c d kc kd ] 4 1 (1) Example [ ] = [ (4) ] = [ (7) ( 2) 21 6 ] Examples: If R = [ ], find 5R If T = [ ], find 4T 9 If A = [ ] and B = [ 4 8 ], find 4B A 2 Upper Level 5 If A = [ ] and B = [ ], find 6B + 7A 7

54 Ns ber Multiplying Matrices 0.20 Topic: Multiplying Matrices Date: Objectives: SWBAT (Multiply Matrices) Main Ideas: Multiplying Matrices Assignment: You can multiply two matrices A and B if and only if the number of columns in A is equal to the number of rows in B. The resulting answer s dimensions will be the rows in A by the columns in B. A B = AB m x r r x t m x t Same dimensions of AB Simple Examples Determine whether each matrix product is defined. If so, state the dimension of the product. A x4 and B 4x2 A 5x and B 5x4 A 4x6 and B 6x2 A x2 and B x2 Multiplying? s A B = AB a b c [ d e f ] g j = ag + bh + ci aj + bk + cl [ h k] [[ dg + eh + fi dj + ek + fl ]] i l Examples: [ ] [ 8 4 1] 5 2 Find UV if U = [ ] and V = [ ]

55 Commutative Property Find each product if G = [ ] and H = [ 2 8] 1 7 GH HG Find each product if J = [ ], K = [ ], and L = [ ] J(K + L) JK + JL Distributive Property

56 Ns ber Solving Systems w/matrices 0.21 Topic: Solving Systems with Matrices Date: Objectives: SWBAT (Solve Systems with the use of Matrix Equation) Main Ideas: Determinant of a Matrix Assignment: Every square matrix has a determinant. A 2 x 2 matrix has a determinant called a second-order determinant. Determinant are denoted by lines around matrix instead of brackets To find it: det [ a b c d ] = a b = ad bc (difference of the products of its diagonals c d c Example: 4 5 = 4(6) 5( ) = = 9 6 Your Turn Determinants of x matrices are called third-order determinants. You find them by using the diagonal rule. Steps for Diagonal Rule: 1. Rewrite the first two columns to the right of the determinant 2. Draw diagonals, beginning with the upper-left element. Multiply the elements in each diagonal. Repeat this process, beginning with the upper right-hand element.. Find the sum of the products of the elements in each set of diagonals. 4. Subtract the second sum from the first sum. x 4 8 Evaluate [ 2 6] using the diagonals 4 5 9

57 Your Turn Inverse of a Matrix The inverse of matrix A = [ a b c d ] is A 1 = 1 [ d ad bc c b ], where ad bc 0. a Examples: Find the inverse of each matrix, if it exists. P = [ ] Q = [ ] The identity matrix is a square matrix that, when multiplied by another matrix, equals the same matrix. 2 x 2 Identity Matrix x Identity Matrix Identity Matrix I = [ ] I = [ 0 1 0] When you multiply a matrix and its inverse, the product is their identity matrix. Examples: Determine whether the matrices are inverses of each other. 1 A = [ 4 2 ] and B = [ ] F = [ ] and G = [ ] 8

58 Solving Systems w/matrices 0.21 Matrices can be used to represent and solve systems of equations. You can use what is called the matrix equation to solve the system of equations below: x + y = 9 x + y [ x 6y = x 6y ] = [9 ] Matrix Equations A X = B 1 2 [ 6 ] [x y ] = [9 ] coefficient matrix variable matrix constant martix Solve matrix equation in the same way you would solve other linear equations. AX = B A 1 AX = B A 1 1X = A 1 B X = A 1 B Answer to the system is the product of constant matrix and the inverse of coefficient matrix. x + y = 2x + y = 6 x + y = 4 x + y = 4 Your Turn

59 Ns ber Guassian Elimination Method 0.22 Topic: Solving Systems with Matrices Date: Objectives: SWBAT (Solve Systems with the use of Guassian Elimination Method) Main Ideas: Elementary Row Operations Assignment: A system of equations can be solved by writing the system in matrix form (like the one below called an Augmented Matrix) and then performing, what is called, elementary row operations on the matrix similar to those performed on equations in the elimination method. System of Equations x 2y + z = 2 x z = 2 2x y + 4z = 5 Augmented Matrix [ ] 5 Elementary Row Operations: 1. Interchange two rows. 2. Multiply all the elements in a row by the same nonzero number.. Replace a row by the sum of that row and a multiple of any other row. Examples of notations: R 1 R 2 Means to inerchange rows 1 and 2 R 2 Means to multiply row 2 by 2R 1 + R 2 R 2 Means to replace row 2 by the sum of that row and 2 times row 1 The ultimate goal is to use elementary row operations to rewrite the augmented matrix with 1 s down the main diagonal and 0 s to the left of the 1 s in each row except the first. Getting into this form (like the one below) is called row echelon form of a matrix. (Guassian Elimination) Row Echelon Form 1 [ ] 2 Then we will use backward substitution to find the solution to the system. Find the solution to the following system using row echelon form and backward substitution. x 6y = 12 { 2x + y =

60 Your Turn Find the solution to the following system using row echelon form and backward substitution (Guassian Elimination). x y = 8 { x y = 0 Find the solution to the following system using row echelon form and backward substitution (Guassian Elimination). x + 2y + 2z = 1 { 4x 10y z = x + 4y + 2z = 2 Variables

61 Guassian Elimination Method 0.22 Find the solution to the following system using row echelon form and backward substitution (Guassian Elimination). 2x + 6y z = { x 2y + 2z = 1 x 6y + 7z = 6 Your Turn Find the solution to the following system using row echelon form and backward substitution (Guassian Elimination). 2x + 6y + 10z = { x + 8y + 15z = 0 x + 2y + z = 1