3.8. F d 2b provided ad 2 cb Þ 0. F d 2b 5 F Use Inverse Matrices to Solve Linear Systems. For Your Notebook E XAMPLE 1
|
|
- Josephine Parsons
- 5 years ago
- Views:
Transcription
1 TEKS 3.8 2A.2.A, 2A.3.A, 2A.3.B, 2A.3.C Use Inverse Matrices to Solve Linear Systems Before You solved linear systems using Cramer s rule. Now You will solve linear systems using inverse matrices. Why? So you can find how many batches of a recipe to make, as in Ex. 45. Key Vocabulary identity matrix inverse matrices matrix of variables matrix of constants The n 3 n identity matrix is a matrix with s on the main diagonal and 0 s elsewhere. If A is any n 3 n matrix and I is the n 3 n identity matrix, then AI 5 A and IA 5 A Identity Matrix Identity Matrix I I G G 0 0 Two n 3 n matrices A and B are inverses of each other if their product (in both orders) is the n 3 n identity matrix. That is, AB 5 I and BA 5 I. An n 3 n matrix A has an inverse if and only if det A Þ 0. The symbol for the inverse of A is A 2. KEY CONCEPT or Your Notebook The Inverse of a Matrix The inverse of the matrix A 5 a c dg b is A 2 5} A d 2b 5 } 2c ag ad 2 cb d 2b provided ad 2 cb Þ 0. 2c ag E XAMPLE ind the inverse of a matrix CHECK INVERSES In Example, you can check the inverse by showing that AA 2 5 I 5 A 2 A. ind the inverse of A G. A 2 5 } G G G GUIDED PRACTICE for Example ind the inverse of the matrix G G 2 22G 20 Chapter 3 Linear Systems and Matrices
2 E XAMPLE 2 Solve a matrix equation Solve the matrix equation AX 5 B for the matrix X. Solution A B GX G Begin by finding the inverse of A. A 2 5} G G 4 7 2G GX G 22 3 A 2 22G 2 AX 5 A 2 B 0 0 GX IX 5 A 3 2G 2 B To solve the equation for X, multiply both sides of the equation by A 2 on the left. at classzone.com X G X 5 A 2 B GUIDED PRACTICE for Example 2 4. Solve the matrix equation GX G. INVERSE O A MATRIX The inverse of a matrix is difficult to compute by hand. A calculator that will compute inverse matrices is useful in this case. E XAMPLE 3 Use a graphing calculator to find the inverse of A. Then use the calculator to verify your result. Solution ind the inverse of a matrix A G Enter matrix A into a graphing calculator and calculate A 2. Then compute AA 2 and A 2 A to verify that you obtain the identity matrix. [A]- [[2-7 3 ] [ ] [.5 -.5]] [A][A]- [[ 0 0] [0 0] [0 0 ]] [A]-[A] [[ 0 0] [0 0] [0 0 ]] 3.8 Use Inverse Matrices to Solve Linear Systems 2
3 GUIDED PRACTICE for Example 3 Use a graphing calculator to find the inverse of the matrix A. Check the result by showing 5 that AA 2 5 I and A 2 A 5 I A A G 2G 8G A KEY CONCEPT or Your Notebook Using an Inverse Matrix to Solve a Linear System STEP Write the system as a matrix equation AX 5 B. The matrix A is the coefficient matrix, X is the matrix of variables, and B is the matrix of constants. STEP 2 ind the inverse of matrix A. STEP 3 Multiply each side of AX 5 B by A 2 on the left to find the solution X 5 A 2 B. E XAMPLE 4 Solve a linear system Use an inverse matrix to solve the linear system. 2x 2 3y 5 9 Equation x 4y 527 Equation 2 SOLVE SYSTEMS You can use the method shown in Example 4 if A has an inverse. If A does not have an inverse, then the system has either no solution or infinitely many solutions. Solution STEP Write the linear system as a matrix equation AX 5 B. coefficient matrix of matrix of matrix (A) variables (X) constants (B) 2 23 p 4G yg x 5 27G 9 STEP 2 ind the inverse of matrix A. A 2 5 } 8 2 (23) G } 2} 2 }4 STEP 3 Multiply the matrix of constants by A 2 on the left. 4 } 3 } 53 X 5 A 2 B 2} 2 }4 9 27G5 5 23G 5 x yg c The solution of the system is (5, 23). CHECK 2(5) 2 3(23) (23) at classzone.com 22 Chapter 3 Linear Systems and Matrices
4 E XAMPLE 5 TAKS REASONING: Multi-Step Problem GITS A company sells three types of movie gift baskets. A basic basket with 2 movie passes and package of microwave popcorn costs $5.50. A medium basket with 2 movie passes, 2 packages of popcorn, and DVD costs $37. A super basket with 4 movie passes, 3 packages of popcorn, and 2 DVDs costs $ ind the cost of each item in the gift baskets. ANOTHER WAY or an alternative method for solving the problem in Example 5, turn to page 28 for the Problem Solving Workshop. Solution STEP Write verbal models for the situation. 2 p 2 p movie pass movie pass 2 p popcorn 5 popcorn basic basket DVD 5 medium basket Equation Equation 2 4 p movie pass 3 p popcorn 2 p DVD 5 super basket Equation 3 STEP 2 STEP 3 Write a system of equations. Let m be the cost of a movie pass, p be the cost of a package of popcorn, and d be the cost of a DVD. 2m p Equation 2m 2p d Equation 2 4m 3p 2d Equation 3 Rewrite the system as a matrix equation p dg G 4 3 2Gm STEP 4 Enter the coefficient matrix A and the matrix of constants B into a graphing calculator. Then find the solution X 5 A 2 B. MATRIX[A] 333 [2 0] [2 2 ] [4 3 2] MATRIX[B] 33 [5.5] [37 ] [72.5] [A]-[B] [[7 ] [.5] [20 ]] 3,3=2 3,=72.5 c A movie pass costs $7, a package of popcorn costs $.50, and a DVD costs $20. GUIDED PRACTICE for Examples 4 and 5 Use an inverse matrix to solve the linear system. 8. 4x y x 2 y x 2 y 525 3x 5y 52 6x 2 3y x 2y 5 8. WHAT I In Example 5, how does the answer change if a basic basket costs $7, a medium basket costs $35, and a super basket costs $69? 3.8 Use Inverse Matrices to Solve Linear Systems 23
5 3.8 EXERCISES HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Exs. 3, 25, and 47 5 TAKS PRACTICE AND REASONING Exs. 2, 34, 4, 46, 5, 52, and 53 5 MULTIPLE REPRESENTATIONS Ex. 45 SKILL PRACTICE. VOCABULARY Identify the matrix of variables and the matrix of constants in the matrix equation GyG 5 22G x 4 2. WRITING Explain how to find the inverse of a matrix A where det A Þ 0. EXAMPLE on p. 20 for Exs. 3 2 EXAMPLE 2 on p. 2 for Exs. 3 8 EXAMPLE 3 on p. 2 for Exs INDING INVERSES ind the inverse of the matrix G G G ERROR ANALYSIS Describe and correct the error in finding the inverse of the matrix 2 4 5G G 2. TAKS REASONING What is the inverse of the matrix G 2 3G 26 30G G? A G B G C 23 SOLVING EQUATIONS Solve the matrix equation. 4 5 }3 }6 24 2G 0 23 D 3 20G 3 2G GX G GX G GX G GX G GX G GX INDING INVERSES Use a graphing calculator to find the inverse of matrix A. Check the result by showing that AA 2 5 I and A 2 A 5 I. 5 0G 4G A A A 3 4 0G A G G G 2G A A G 5 6G G 24 Chapter 3 Linear Systems and Matrices
6 EXAMPLE 4 on p. 22 for Exs SYSTEMS O TWO EQUATIONS Use an inverse matrix to solve the linear system x 2 y x 7y x 2 2y x 2 2y x 3y 524 6x 2 5y x 2 y x 2 9y x 2 7y 526 9x 2 0y x 6y 5 8 2x 5y x y x y x 7y x 3y x 5y x 5y TAKS REASONING What is the solution 3x 2 5y 5226 of the system shown? 2x 2y 5 0 A (3, 7) B (7, 2) C (22, 4) D (68, 0) EXAMPLE 5 on p. 23 for Exs SYSTEMS O THREE EQUATIONS Use an inverse matrix and a graphing calculator to solve the linear system. 35. x 2 y 2 3z x y 2 8z x 4y 5z 5 5 5x 2y z 527 x 2 2y z 52 x 2y 3z x 2 y 5 8 2x 2 2y 5z 527 5x 2 4y 2 2z x 2 y 2 z x 2y 2 z x y 2z 5 6x 2 z x 2 5y 4z 5248 x 2 y z 525 2x 4y 5z x y z 5 2 2x 4y 2 z TAKS REASONING Write a matrix that has no inverse. 42. CHALLENGE Solve the linear system using the given inverse of the coefficient matrix. G 2w 5x 2 4y 6z x y 2 7z w 8x 2 7y 4z 5225 A w 6x 2 5y 0z PROBLEM SOLVING EXAMPLES 4 and 5 on pp for Exs AVIATION A pilot has 200 hours of flight time in single-engine airplanes and twin-engine airplanes. Renting a single-engine airplane costs $60 per hour, and renting a twin-engine airplane costs $240 per hour. The pilot has spent $2,000 on airplane rentals. Use an inverse matrix to find how many hours the pilot has flown each type of airplane. 44. BASKETBALL During the NBA season, Dirk Nowitzki of the Dallas Mavericks made a total of 976 shots and scored 680 points. His shots consisted of 3-point field goals, 2-point field goals, and -point free throws. He made 35 more 2-point field goals than free throws. Use an inverse matrix to find how many of each type of shot he made. 3.8 Use Inverse Matrices to Solve Linear Systems 25
7 45. MULTIPLE REPRESENTATIONS A cooking class wants to use up 8 cups of buttermilk and eggs by baking rolls and muffins to freeze. A batch of rolls uses 2 cups of buttermilk and 3 eggs. A batch of muffins uses cup of buttermilk and egg. a. Writing a System Write a system of equations for this situation. b. Writing a Matrix Equation Write the system of equations from part (a) as a matrix equation AX 5 B. c. Solving a System Use an inverse matrix to solve the system of equations. How many batches of each recipe should the class make? 46. TAKS REASONING A company sells party platters with varying assortments of meats and cheeses. A basic platter with 2 cheeses and 3 meats costs $8, a medium platter with 3 cheeses and 5 meats costs $28, and a super platter with 7 cheeses and 0 meats costs $60. a. Write and solve a system of equations using the information about the basic platter and the medium platter. b. Write and solve a system of equations using the information about the medium platter and the super platter. c. Compare the results from parts (a) and (b) and make a conjecture about why there is a discrepancy. 47. NUTRITION The table shows the calories, fat, and carbohydrates per ounce for three brands of cereal. How many ounces of each brand should be combined to get 500 calories, 3 grams of fat, and 00 grams of carbohydrates? Round your answers to the nearest tenth of an ounce. Cereal Calories at Carbohydrates Bran Crunchies 78 g 22 g Toasted Oats 04 0 g 25.5 g Whole Wheat lakes g 23.8 g 48. MULTI-STEP PROBLEM You need 9 square feet of glass mosaic tiles to decorate a wall of your kitchen. You want the area of the red tiles to equal the combined area of the yellow and blue tiles. The cost of a sheet of glass tiles having an area of 0.75 square foot is $6.50 for red, $4.50 for yellow, and $8.50 for blue. You have $80 to spend. a. Write a system of equations to represent this situation. b. Rewrite the system as a matrix equation. c. Use an inverse matrix to find how many sheets of each color tile you should buy. 49. GEOMETRY The columns of matrix T below give the coordinates of the vertices of a triangle. Matrix A is a transformation matrix. A 5 0 T 2 0G G a. ind AT and AAT. Then draw the original triangle and the two transformed triangles. What transformation does A represent? b. Describe how to use matrices to obtain the original triangle represented by T from the transformed triangle represented by AAT. Mosaic tiles 26 5 WORKED-OUT SOLUTIONS on p. WS 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS
8 50. CHALLENGE Verify the formula on page 20 for the inverse of a matrix by showing that AB 5 I and BA 5 I for the matrices A and B given below. A 5 a b B 5 } c dg ad 2 cb d 2b 2c ag MIXED REVIEW OR TAKS TAKS PRACTICE at classzone.com REVIEW TAKS Preparation p. 66; TAKS Workbook REVIEW Lesson.3; TAKS Workbook REVIEW Lesson 3.2; TAKS Workbook 5. TAKS PRACTICE A grocer wants to mix peanuts worth $2.50 per pound with 2 pounds of cashews worth $4.75 per pound. To obtain a nut mixture worth $4 per pound, how many pounds of peanuts are needed? TAKS Obj. 0 A 3.6 lb B 6 lb C 2 lb D 8 lb 52. TAKS PRACTICE The sum of three numbers is 4. The second number is 5 less than three times the first number. The third number is 2 more than four times the first number. Which equation represents the relationship between the three numbers where n is the first number? TAKS Obj. 4 5 n 2 (3n 2 5) 2 (4n 2) G 4 5 n (4n 2 5) (3n 2) H 4 5 n (3n 2 5) (4n 2) J 4 5 n (5 2 3n) (2 4n) 53. TAKS PRACTICE Which ordered pair is the solution of this system of linear equations? TAKS Obj. 4 5x y 527 2x 2 7y 5 8 A (23, 22) B (23, 2) C 3, 2 2 } 7 2 D (, 2) QUIZ for Lessons Using the given matrices, evaluate the expression. (p. 95) A 5 24 B 5 2G, C 0 G, 5 2 4G. 2AB 2. AB AC 3. A(B C) 4. (B 2 A)C Evaluate the determinant of the matrix. (p. 203) G G 7. Use an inverse matrix to solve the linear system. (p. 20) 2 G x 3y x 2 4y x 2y 523 2x 7y 526 2x 2 3y 5 3 6x 2 5y x 2 y x 4y x y 522 2x 2 2y 528 5x 3y x y BOATING You are making a triangular sail for a sailboat. The vertices of the sail are (0, 2), (2, 2), and (2, 26) where the coordinates are measured in feet. ind the area of the sail. (p. 203) EXTRA PRACTICE for Lesson 3.8, p. 02 ONLINE QUIZ at classzone.com 27
9 LESSON 3.8 TEKS 2A.3.A, 2A.3.B, 2A.3.C Using ALTERNATIVE METHODS Another Way to Solve Example 5, page 23 MULTIPLE REPRESENTATIONS In Example 5 on page 23, you solved a linear system using an inverse matrix. You can also solve systems using augmented matrices. An augmented matrix for a system contains the system s coefficient matrix and matrix of constants. Linear System x 2 4y x 7y 522 Augmented Matrix G Recall from Lesson 3.2 that an equation in a system can be multiplied by a constant, or a multiple of one equation can be added to another equation. Similar operations can be performed on the rows of an augmented matrix to solve the corresponding system. KEY CONCEPT or Your Notebook Elementary Row Operations for Augmented Matrices Two augmented matrices are row-equivalent if their corresponding systems have the same solution(s). Any of these row operations performed on an augmented matrix will produce a matrix that is row-equivalent to the original: Interchange two rows. Multiply a row by a nonzero constant. Add a multiple of one row to another row. P ROBLEM GITS A company sells three types of movie gift baskets. A basic basket with 2 movie passes and package of microwave popcorn costs $5.50. A medium basket with 2 movie passes, 2 packages of popcorn, and DVD costs $37. A super basket with 4 movie passes, 3 packages of popcorn, and 2 DVDs costs $ ind the cost of each item in the gift baskets. M ETHOD Using an Augmented Matrix You need to write a linear system, write the corresponding augmented matrix, and use row operations to transform the augmented matrix into a matrix with s along the main diagonal and 0 s below the main diagonal. Such a matrix is in triangular form and can be used to solve for the variables in the system. Let m be the cost of a movie pass, p be the cost of a package of popcorn, and d be the cost of a DVD. 28 Chapter 3 Linear Systems and Matrices
10 STEP Write a linear system and then write an augmented matrix. 2m p m 2p d m 3p 2d G STEP 2 Add 22 times the first row to the third row (22)R R G STEP 3 Add 2 times the first row to the second row (2)R R G STEP 4 Add 2 times the second row to the third row (2)R 2 R G STEP 5 Multiply the first row by R G The third row of the matrix tells you that d Substitute 20 for d in the equation for the second row, p d 5 2.5, to obtain p , or p 5.5. Then substitute.5 for p in the equation for the first row, m 0.5p , to obtain m 0.5(.5) , or m 5 7. c A movie pass costs $7, a package of popcorn costs $.50, and a DVD costs $20. P RACTICE. WHAT I? In the problem on page 28, suppose a basic basket costs $7.75, a medium basket costs $34.50, and a super basket costs $ Use an augmented matrix to find the cost of each item. 2. INANCE You have $8,000 to invest. You want an overall annual return of 8%. The expected annual returns are 0% for a stock fund, 7% for a bond fund, and 5% for a money market fund. You want to invest as much in stocks as in bonds and the money market combined. Use an augmented matrix to find how much to invest in each fund. 3. BIRDSEED A pet store sells 20 pounds of birdseed for $0.85. The birdseed is made from two kinds of seeds, sunflower seeds and thistle seeds. Sunflower seeds cost $.34 per pound and thistle seeds cost $.79 per pound. Use an augmented matrix to find how many pounds of each variety are in the mixture. 4. REASONING Solve the given system using an augmented matrix. What can you say about the system s solution(s)? x 2 2y 4z 520 5x y 2 z x 2 6y 2z 5230 Using Alternative Methods 29
Perform Function Operations and Composition
TEKS 6.3 a.3, 2A.1.A Perform Function Operations and Composition Before You performed operations with algebraic epressions. Now You will perform operations with functions. Why? So you can model biological
More informationPractice Test - Chapter 6
1. Write each system of equations in triangular form using Gaussian elimination. Then solve the system. Align the variables on the left side of the equal sign. Eliminate the x-term from the 2nd equation.
More informationWrite Polynomial Functions and Models
TEKS 5.9 a.3, A.1.B, A.3.B; P.3.B Write Polnomial Functions and Models Before You wrote linear and quadratic functions. Now You will write higher-degree polnomial functions. Wh? So ou can model launch
More informationPrecalculus Notes: Unit 7 Systems of Equations and Matrices
Date: 7.1, 7. Solving Systems of Equations: Graphing, Substitution, Elimination Syllabus Objectives: 8.1 The student will solve a given system of equations or system of inequalities. Solution of a System
More informationVerify Trigonometric Identities
4.3 a., A..A; P..C TEKS Verify Trigonometric Identities Before You graphed trigonometric functions. Now You will verify trigonometric identities. Why? So you can model the path of Halley s comet, as in
More information12.5 Investigate the Volume of a Pyramid
Investigating g Geometry ACTIVITY Use before Lesson 12.5 12.5 Investigate the Volume of a Pyramid MATERIALS ruler poster board scissors tape uncooked rice Q U E S T I O N How is the volume of a pyramid
More informationMath 13 Chapter 3 Handout Helene Payne. Name: 1. Assign the value to the variables so that a matrix equality results.
Matrices Name:. Assign the value to the variables so that a matrix equality results. [ [ t + 5 4 5 = 7 6 7 x 3. Are the following matrices equal, why or why not? [ 3 7, 7 4 3 4 3. Let the matrix A be defined
More informationMatrix Inverse 2 ( 2) 1 = 2 1 2
Name: Matrix Inverse For Scalars, we have what is called a multiplicative identity. This means that if we have a scalar number, call it r, then r multiplied by the multiplicative identity equals r. Without
More informationGraph and Write Equations of Hyperbolas
TEKS 9.5 a.5, 2A.5.B, 2A.5.C Graph and Write Equations of Hperbolas Before You graphed and wrote equations of parabolas, circles, and ellipses. Now You will graph and write equations of hperbolas. Wh?
More informationFind Rational Zeros. has integer coefficients, then every rational zero of f has the following form: x 1 a 0. } 5 factor of constant term a 0
.6 Find Rational Zeros TEKS A.8.B; P..D, P..A, P..B Before You found the zeros of a polnomial function given one zero. Now You will find all real zeros of a polnomial function. Wh? So ou can model manufacturing
More informationFinite Math - J-term Homework. Section Inverse of a Square Matrix
Section.5-77, 78, 79, 80 Finite Math - J-term 017 Lecture Notes - 1/19/017 Homework Section.6-9, 1, 1, 15, 17, 18, 1, 6, 9, 3, 37, 39, 1,, 5, 6, 55 Section 5.1-9, 11, 1, 13, 1, 17, 9, 30 Section.5 - Inverse
More informationLesson 6.1 Matrix Representations
Lesson. Matrix Representations. Supply the missing entries in each transition matrix..7 m r.9..7 a. [M] b. [R] c. [T] t. m. A survey of registered voters showed that of those people who voted in the presidential
More informationGraph General Rational Functions. }} q(x) bn x n 1 b n 2 1. p(x) 5 a m x m 1 a m 2 1
TEKS 8.3 A.0.A, A.0.B, A.0.C, A.0.F Graph General Rational Functions Before You graphed rational functions involving linear polnomials. Now You will graph rational functions with higher-degree polnomials.
More informationx = 12 x = 12 1x = 16
2.2 - The Inverse of a Matrix We've seen how to add matrices, multiply them by scalars, subtract them, and multiply one matrix by another. The question naturally arises: Can we divide one matrix by another?
More informationExponents and Real Numbers
Exponents and Real Numbers MODULE? ESSENTIAL QUESTION What sets of numbers are included in the real numbers? CALIFORNIA COMMON CORE LESSON.1 Radicals and Rational Exponents N.RN.1, N.RN. LESSON. Real Numbers
More information0_PreCNotes17 18.notebook May 16, Chapter 12
Chapter 12 Notes BASIC MATRIX OPERATIONS Matrix (plural: Matrices) an n x m array of elements element a ij Example 1 a 21 = a 13 = Multiply Matrix by a Scalar Distribute scalar to all elements Addition
More informationEvaluate and Graph Polynomial Functions
5.2 Evaluate and Graph Polnomial Functions Before You evaluated and graphed linear and quadratic functions. Now You will evaluate and graph other polnomial functions. Wh? So ou can model skateboarding
More information8 Quadrilaterals. Before
8 Quadrilaterals 8. Find Angle Measures in Polygons 8. Use Properties of Parallelograms 8.3 Show that a Quadrilateral is a Parallelogram 8.4 Properties of Rhombuses, Rectangles, and Squares 8.5 Use Properties
More information3. Replace any row by the sum of that row and a constant multiple of any other row.
Math Section. Section.: Solving Systems of Linear Equations Using Matrices As you may recall from College Algebra or Section., you can solve a system of linear equations in two variables easily by applying
More information1. Expand and simplify: 9. Determine the area of the triangle. a. 5 b. 2. Simplify: a. ( 3) c. ( 2) d. 4 ( 2) 3. e ( 3) ( 5)
Math 08 Final Exam Review. Expand and simplify: (Section.6). Simplify: ( ) ( ) 00 d. ( ) (Section.,.6,.) (Section.,.,.) 6 ( ) ( ) e. 6 6 8 (Section.,.6,.) (Section.,.) ( 6 ) (Section.,.,.). Translate into
More informationExercise Set Decide whether each matrix below is an elementary matrix. (a) (b) (c) (d) Answer:
Understand the relationships between statements that are equivalent to the invertibility of a square matrix (Theorem 1.5.3). Use the inversion algorithm to find the inverse of an invertible matrix. Express
More information2-2. Solving Systems of Equations in Three Variables. OBJECTIVE Solve systems of equations involving three variables algebraically.
2-2 OBJECTIVE Solve systems of equations involving three variables algebraically. You will learn more about graphing in threedimensional space in Chapter 8. Solving Systems of Equations in Three Variables
More informationCollege Algebra Exam File - Fall Test #1
College Algebra Exam File - Fall 010 Test #1 1.) For each of the following graphs, indicate (/) whether it is the graph of a function and if so, whether it the graph of one-to one function. Circle your
More informationSurface Area and Volume
8 Surface Area and Volume 8. Three-Dimensional Figures 8. Surface Areas of Prisms 8. Surface Areas of Pyramids 8. Volumes of Rectangular Prisms I petitioned my owner for a doghouse with greater volume.
More informationReview for Elementary Algebra Final Exam
Review for Elementary Algebra Final Exam The quadratic formula will be provided on the final exam. Students are expected to know from memory all other relevant formulas, including: - Sum of the angles
More information12.2 Investigate Surface Area
Investigating g Geometry ACTIVITY Use before Lesson 12.2 12.2 Investigate Surface Area MATERIALS grap paper scissors tape Q U E S T I O N How can you find te surface area of a polyedron? A net is a pattern
More information2. Find the measure of exterior angle. 3. Find the measures of angles A, B, and C. 4. Solve for x. 5. Find the measure of
INTEGRATED MATH III SUMMER PACKET DUE THE FIRST DAY OF SCHOOL The problems in this packet are designed to help you review topics from previous mathematics courses that are essential to your success in
More information16.3 Volume of Cones
Name Class Date 16. Volume of Cones Essential Question: How do you calculate the volumes of composite figures that include cones? Explore G.11.D Apply the formulas for the volume of three-dimensional figures,
More informationFor example, the system. 22 may be represented by the augmented matrix
Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural of matrix) may be
More informationWhat is a tessellation???? Give an example... Daily Do from last class Homework Answers 10 7 These are similar: What does y =? x =?
Daily Do from last class Homework Answers 10 7 These are similar: What does y =? x =? 36 74 0 78 0 154 o 44 48 54 o y x 154 o 78 0 12 74 0 9 1. 8 ft 2. 21m 3. 21 ft 4. 30cm 5. 6mm 6. 16 in 7. yes 9 = 7
More informationIndependent systems consist of x
5.1 Simultaneous Linear Equations In consistent equations, *Find the solution to each system by graphing. 1. y Independent systems consist of x Three Cases: A. consistent and independent 2. y B. inconsistent
More informationPart 1: Dividing Fractions Using Visual Representations
Part : Dividing Fractions Using Visual Representations To divide fractions, remember that division can be represented by repeated subtraction, just like multiplication can be represented by repeated addition.
More informationChapter 1 & 2. Homework Ch 1 & 2
Chapter 1 & 2 1-1 Relations & Functions 1-2 Compostion of Functions 1-3 Graphs Linear Eqns 1-4 Writing Linear Functions 1-5 Parallel & Perpendicular Lines 1-7 Piecewise Functions 1-8 Linear Inequalities
More informationMA 162: Finite Mathematics - Sections 2.6
MA 162: Finite Mathematics - Sections 2.6 Fall 2014 Ray Kremer University of Kentucky September 24, 2014 Announcements: Homework 2.6 due next Tuesday at 6pm. Multiplicative Inverses If a is a non-zero
More informationSystems of Inequalities and Linear Programming 5.7 Properties of Matrices 5.8 Matrix Inverses
5 5 Systems and Matrices Systems and Matrices 5.6 Systems of Inequalities and Linear Programming 5.7 Properties of Matrices 5.8 Matrix Inverses Sections 5.6 5.8 2008 Pearson Addison-Wesley. All rights
More informationMath 112 Fall 2014 Midterm 1 Review Problems Page 1. (E) None of these
Math Fall Midterm Review Problems Page. Solve the equation. The answer is: x x 7 Less than Between and Between and Between and 7 (E) More than 7. Solve for x : x x 8. The solution is a number: less than
More informationVolume of Spheres. A geometric plane passing through the center of a sphere divides it into. into the Northern Hemisphere and the Southern Hemisphere.
9.6 Surface Area and Volume of Spheres Goal Find surface areas and volumes of spheres. Key Words sphere hemisphere A globe is an example of a sphere. A sphere is the set of all points in space that are
More informationEXTENSION. a 1 b 1 c 1 d 1. Rows l a 2 b 2 c 2 d 2. a 3 x b 3 y c 3 z d 3. This system can be written in an abbreviated form as
EXTENSION Using Matrix Row Operations to Solve Systems The elimination method used to solve systems introduced in the previous section can be streamlined into a systematic method by using matrices (singular:
More information5th Grade Mathematics Essential Standards
Standard 1 Number Sense (10-20% of ISTEP/Acuity) Students compute with whole numbers*, decimals, and fractions and understand the relationship among decimals, fractions, and percents. They understand the
More information1. Solve the system by graphing: x y = 2 2. Solve the linear system using any method. 2x + y = -7 2x 6y = 12
1. Solve the system by graphing: x y =. Solve the linear system using any method. x + y = -7 x 6y = 1 x + y = 8 3. Solve the linear system using any method. 4. A total of $0,000 is invested in two funds
More informationSurface Area and Volume of Solids
Surface Area and Volume of Solids? MODULE 5 LESSON 5. ESSENTIAL QUESTION Nets and Surface Area How can a model help you to solve surface area and volume problems? 6.G. LESSON 5. Volume of Rectangular Prisms
More informationPoe Middle School Rising 8 th Grade Geometry Students Summer Mathematics Packet
Poe Middle School Rising 8 th Grade Geometry Students Summer Mathematics Packet Dear Students: The purpose of this packet is to review Algebra concepts as you look forward to Geometry at Poe Middle School,
More informationSolutions Key Matrices
CHAPTER Solutions Ke Matrices ARE YOU READY? PAGE. B. E. C. D.. 7. 8. 9..........7 7. 8 8. 9 9. JKL and MNP. CAB. GH KL FH JL KL 8 8KL KL 8 MATRICES AND DATA, PAGES CHECK IT OUT! a. b. c. m and m a. B
More informationExplore Solids
1212.1 Explore Solids Surface Area and Volume of Solids 12.2 Surface Area of Prisms and Cylinders 12.3 Surface Area of Pyramids and Cones 12.4 Volume of Prisms and Cylinders 12.5 Volume of Pyramids and
More informationArchdiocese of Washington Catholic Schools Academic Standards Mathematics
5 th GRADE Archdiocese of Washington Catholic Schools Standard 1 - Number Sense Students compute with whole numbers*, decimals, and fractions and understand the relationship among decimals, fractions,
More informationLesson 10: Representing, Naming, and Evaluating Functions
: Representing, Naming, and Evaluation Functions Classwork Opening Exercise Study the 4 representations of a function below. How are these representations alike? How are they different? TABLE: Input 0
More information2-9 Operations with Complex Numbers
2-9 Operations with Complex Numbers Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Express each number in terms of i. 1. 9i 2. Find each complex conjugate. 3. 4. Find each product. 5. 6. Objective
More informationCHAPTER 5 SYSTEMS OF EQUATIONS. x y
page 1 of Section 5.1 CHAPTER 5 SYSTEMS OF EQUATIONS SECTION 5.1 GAUSSIAN ELIMINATION matrix form of a system of equations The system 2x + 3y + 4z 1 5x + y + 7z 2 can be written as Ax where b 2 3 4 A [
More informationSemester 2 Review Units 4, 5, and 6
Precalculus Semester 2 Review Units 4, 5, and 6 NAME: Period: UNIT 4 Simplify each expression. 1) (sec θ tan θ)(1 + tan θ) 2) cos θ sin 2 θ 1 3) 1+tan θ 1+cot θ 4) cos 2θ cosθ sin θ 5) sec 2 x sec 2 x
More informationRational Numbers pp
LESSON 3-1 Rational Numbers pp. 112 114 Vocabulary rational number (p. 112) relatively prime (p. 112) Additional Examples Example 1 Simplify. 5 1 0 5 1 5 is a common factor. 10 2 5 5 1 5 Divide the numerator
More information17.2 Surface Area of Prisms
h a b c h a b c Locker LESSON 17. Surface Area of Prisms and Cylinders Texas Math Standards The student is expected to: G.11.C Apply the formulas for the total and lateral surface area of three-dimensional
More information10/26/ Solving Systems of Linear Equations Using Matrices. Objectives. Matrices
6.1 Solving Systems of Linear Equations Using Matrices Objectives Write the augmented matrix for a linear system. Perform matrix row operations. Use matrices and Gaussian elimination to solve systems.
More informationGrade 6: Geometry and Algebra. 1. Choose an appropriate customary unit for measuring the area of a classroom. B. Square Feet D.
6.8.1 Name Grade 6: Geometry and Algebra 1. Choose an appropriate customary unit for measuring the area of a classroom. A. Cubic Feet C. Square Inches B. Square Feet D. Square Acres 2. What is the measure
More informationTo add or subtract, just add or subtract the numbers in the same column and row and place answer accordingly.
Math 3 Variable Manipulation Part 2 Systems with Matrices MATRICES An alternative method to solving system of equations is using Matrices. However, before we can solve systems of equations using matrices,
More informationGraphing Linear Equations Review
Graph each equation. Graphing Linear Equations Review 1. y = 3x 1 2. y = 1 x 2 3 2 3. 6x 3y 12 4. y x 4 5 What do we know about parallel and perpendicular lines? Parallel lines Perpendicular lines What
More informationLesson 19. Opening Discussion
Opening Discussion 1. Think about the forms of the quadratic equations you ve written throughout this module. We have gone from vertex form to standard form and from factored form to standard form. Draw
More informationCommon Core. Mathematics Instruction
0 Common Core Mathematics Instruction 8 Table of Contents Unit : Expressions and Equations (Exponents) and the Number System.............................. Lesson Properties of Integer Exponents....................
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3: Systems of Equations Mrs. Leahy 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system of equations
More informationMath B Regents Exam 0603 Page b For which value of x is y = log x undefined?
Math B Regents Exam 0603 Page 1 1. 060301b For which value of x is y = log x undefined? [A] 1 10. 06030b, P.I. A.A.58 [B] 1483. [C] π [D] 0 If sinθ > 0 and sec θ < 0, in which quadrant does the terminal
More information6-2 Matrix Multiplication, Inverses and Determinants
Find AB and BA, if possible. 4. A = B = A = ; B = A is a 2 1 matrix and B is a 1 4 matrix. Because the number of columns of A is equal to the number of rows of B, AB exists. To find the first entry of
More informationUnit 5 Applying Similarity of Triangles
Unit 5 Applying Similarity of Triangles Lesson 1: Proof of the Triangle Side Splitter Theorem Opening Exercise We are going to construct a proof designed to demonstrate the following theorem: A line segment
More informationTest Booklet. Subject: MA, Grade: 10 TAKS Grade 10 Math Student name:
Test Booklet Subject: MA, Grade: 10 TAKS Grade 10 Math 2009 Student name: Author: Texas District: Texas Released Tests Printed: Saturday July 14, 2012 1 The grid below shows the top view of a 3-dimensional
More information, 6.7,, Order the numbers from least to greatest. 1. 1, 0, 2, 5, 4. Simplify the expression. 10.
Getting Ready for Pre-Algebra or Algebra Summer Math Practice The following are practice questions to evaluate the students understanding of concepts and skills taught in seventh grade as a readiness for
More informationCourse Number 432/433 Title Algebra II (A & B) H Grade # of Days 120
Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number
More informationLesson 3.3. Multiplication of Matrices, Part 2
Lesson 3.3 Multiplication of Matrices, Part 2 In Lesson 3.2 you multiplied a row matrix times a 2 by 3 price matrix to determine the total cost of 5 pizzas and 3 salads at each pizza house. (See page 133.)
More informationThe Iranian tile pattern at the right shows several polygons. These polygons can be classified as convex or concave. interior
Page 1 of 5 8.1 Classifying Polygons Goal Describe polygons. Key Words convex concave equilateral equiangular regular The Iranian tile pattern at the right shows several polygons. These polygons can be
More information1. How would you describe the relationship between the x- and y-values in the scatter plot?
Unit 2 Test Review Name Per 1. How would you describe the relationship between the x- and y-values in the scatter plot? 90 80 70 60 50 0 '90 '95 '00 '05 '10 2. Based on the data in the scatter plot in
More informationDATE PERIOD. Lesson Reading Guide
NAME DATE PERIOD Lesson Reading Guide Get Ready for the Lesson Read the introduction at the top of page 316 in your textbook. Write your answers below. 1. Predict the number of triangles and the sum of
More informationEXAMPLE: Multiplying Integers with the Same Sign. Recall that multiplication is repeated addition. 3 2 means to add 3 groups of 2.
1.4 Multiplying Integers COMMON CORE STATE STANDARDS 7.EE. 7.NS.2a 7.NS.2c negative, or zero? How can you tell? 1 Use repeated addition to find 2. Is the product of two integers positive, : Multiplying
More informationChapter 1: Variables, Expressions, and Integers
Name: Pre-Algebra Period: 8 Chapter 1: Variables, Expressions, and Integers Outline 1.1: p. 7 #12-15, 20-27, 32, 33, 34, 36 Date 1.2: p. 12 #16-20, 25-28, 30, 31, 36 1.3: p. 19 #10-18, 21-25, 31 1.4: p.
More informationConverse of the. Pythagorean Theorem 8.7.C. Decide whether the converse of the Pythagorean Theorem is true.
LESSON 8. Converse of the Pythagorean Theorem ESSENTIAL QUESTION Expressions, equations, and relationships Use the Pythagorean Theorem and its converse to solve problems. How can you test the converse
More information1 of 39 8/14/2018, 9:48 AM
1 of 39 8/14/018, 9:48 AM Student: Date: Instructor: Alfredo Alvarez Course: Math 0410 Spring 018 Assignment: Math 0410 Homework150bbbbtsiallnew 1. Graph each integer in the list on the same number line.
More informationLesson 10: Interpreting Graphs of Proportional Relationships
Classwork Example 1 Grandma s Special Chocolate Chip Cookie recipe, which yields 4 dozen cookies, calls for 3 cups of flour. Using this information, complete the chart: Create a table comparing the amount
More informationBUILD YOUR VOCABULARY
C H A P T E R 12 BUILD YOUR VOCABULARY This is an alphabetical list of new vocabulary terms you will learn in Chapter 12. As you complete the study notes for the chapter, you will see Build Your Vocabulary
More informationName Class Date. Using Graphs to Relate Two Quantities
4-1 Reteaching Using Graphs to Relate Two Quantities An important life skill is to be able to a read graph. When looking at a graph, you should check the title, the labels on the axes, and the general
More informationChapter 3 Transformations of Graphs and Data
Chapter 3 Transformations of Graphs and Data 3.1 Graphs of Parent Functions Parent Function the simplest equation of a particular type of function o Ex: Quadratic Function: y = x 2 There are EIGHT important
More informationReady To Go On? Skills Intervention 10-1 Solid Geometry
10A Find these vocabulary words in Lesson 10-1 and the Multilingual Glossary. Vocabulary Ready To Go On? Skills Intervention 10-1 Solid Geometry face edge vertex prism cylinder pyramid cone cube net cross
More informationMath 20 Practice Exam #2 Problems and Their Solutions!
Math 20 Practice Exam #2 Problems and Their Solutions! #1) Solve the linear system by graphing: Isolate for in both equations. Graph the two lines using the slope-intercept method. The two lines intersect
More information_ _ _ _ _ _. Add and Subtract Parts of a Whole R54. Name. Justin has 3_. How much cheese does he have in all?
Lesson 7.1 Add and Subtract Parts of a Whole Justin has 8pound of cheddar cheese and 2_ 8pound of brick cheese. How much cheese does he have in all? Step 1 Use fraction strips to model the problem. Use
More informationLesson 6: Using Matrices to Solve Systems of Equations
Opening Exercise In Module 1, ou learned to solve sstems of equations b graphing, and b the substitution and elimination methods. Sstems of equations can also be solved using matrices. Before we use matrices,
More informationExcel for Algebra 1 Lesson 5: The Solver
Excel for Algebra 1 Lesson 5: The Solver OK, what s The Solver? Speaking very informally, the Solver is like Goal Seek on steroids. It s a lot more powerful, but it s also more challenging to control.
More informationDo Now: For the following pair of similar figures, write the ratio of side lengths
CC Geometry H Aim #8: What is the relationship between the ratio of the side lengths of similar solids and the ratio of their volumes? Do Now: For the following pair of similar figures, write the ratio
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 1324 Final Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the system of two equations in two variables. 1) 8x + 9y = -20 3x +
More informationDo you need a worksheet or a copy of the teacher notes? Go to
Name Period Day Date Assignment (Due the next class meeting) Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday
More informationWarm-Up Up Exercises. Use this diagram for Exercises If PR = 12 and m R = 19, find p. ANSWER If m P = 58 and r = 5, find p.
Warm-Up Up Exercises Use this diagram for Exercises 1 4. 1. If PR = 12 and m R = 19, find p. ANSWER 11.3 2. If m P = 58 and r = 5, find p. ANSWER 8.0 Warm-Up Up Exercises Use this diagram for Exercises
More informationAcute Angle. Angle. An angle that measures greater than 0 and less than 90. A figure formed by two line segments or rays that share the same endpoint.
Acute Angle An angle that measures greater than 0 and less than 90. Geometry Angle A figure formed by two line segments or rays that share the same endpoint. 5 Geometry Area The number of square units
More informationFor Exercises 2 4, find the volume and the surface area of each closed box.
Applications 1. a. Make a sketch of an open 1-3-5 box. Label the edges of the box. b. Sketch three boxes that have twice the volume of a 1-3-5 box. Label each box with its dimensions. c. Are any of the
More informationAH Matrices.notebook November 28, 2016
Matrices Numbers are put into arrays to help with multiplication, division etc. A Matrix (matrices pl.) is a rectangular array of numbers arranged in rows and columns. Matrices If there are m rows and
More informationAdd and Subtract Parts of a Whole
Lesson 7. Add and Subtract Parts of a Whole Justin has _ pound of cheddar cheese and 2_ pound of brick cheese. How much cheese does he have in all? Step Use fraction strips to model the problem. Use three
More informationEngage NY Lesson 15: Representing Three-Dimensional Figures Using Nets
Name: Surface Area & Volume Packet Engage NY Lesson 15: Representing Three-Dimensional Figures Using Nets Classwork Cereal Box Similarities: Cereal Box Differences: Exercise 1 1. Some of the drawings below
More information1-3 Multiplying and Dividing Real Numbers
Multiplying and Dividing 1-3 Multiplying and Dividing Real Numbers Real Numbers Warm Up Lesson Presentation Lesson Quiz 1 2 pts Bell Quiz 1-3 Add or Subtract 1. 3 8 2 pts 2. - 8 + 12 2 pts 3. 4 (-4) 2
More informationPre-Calculus Calendar Chapter 1: Linear Relations and Functions Chapter 2: Systems of Linear Equations and Inequalities
Pre-Calculus Calendar Chapter : Linear Relations and Functions Chapter : Systems of Linear Equations and Inequalities /9 Monday Tuesday Wednesday Thursday Friday /0 / / /. p. 0: 9-5,, 5, 7-,, 5, 7-50.
More informationReady To Go On? Skills Intervention 9-1 Developing Formulas for Triangles and Quadrilaterals
9A Ready To Go On? Skills Intervention 9-1 Developing Formulas for Triangles and Quadrilaterals Finding Measurements of Parallelograms Find each measurement. A. the area of the parallelogram A b Use the
More informationMathematics UNIT TWO Transformations and Operations. f -1 (x) f g = f(g(x)) y = af[b(x - h)] + k. Unit. Student Workbook. (f + g)(x) (f - g)(x)
Mathematics 30- Student Workbook Unit 2 Lesson : Basic Transformations Approximate Completion Time: 2 Days y = af[b(x - h)] + k Lesson 2: Combined Transformations Approximate Completion Time: 2 Days f
More informationLinear Equation Systems Iterative Methods
Linear Equation Systems Iterative Methods Content Iterative Methods Jacobi Iterative Method Gauss Seidel Iterative Method Iterative Methods Iterative methods are those that produce a sequence of successive
More informationFor the test, be sure to show all work! PROBLEMS: ANSWERS: For problems 1 9, simplify the expression ( ) Evaluate if x = -2 and y = 1 8.
Pre-algebra For the test, be sure to show all work! PROBLEMS: For problems 9, simplify the expression.. 9 ( 7) Ch / Review.. 56 74... 4. 4 7 4. 5. ( 87) + ( ) 5. 6. 6. 7. ( ) + 4 6 5 + 7. Evaluate if x
More informationIs the product of two integers positive, negative, or zero? How can you tell? ACTIVITY: Multiplying Integers with the Same Sign
1.4 Multiplying Integers Is the product of two integers positive, negative, or zero? How can you tell? 1 ACTIVITY: Multiplying Integers with the Same Sign Work with a partner. Use repeated addition to
More information11-9 Areas of Circles and Sectors. CONSTRUCTION Find the area of each circle. Round to the nearest tenth. 1. Refer to the figure on page 800.
CONSTRUCTION Find the area of each circle. Round to the nearest tenth. 1. Refer to the figure on page 800. Find the indicated measure. Round to the nearest tenth. 3. Find the diameter of a circle with
More informationMississippi College and Career Readiness Standards for Mathematics Scaffolding Document. Grade 4
Mississippi College and Career Readiness Standards for Mathematics Scaffolding Document Grade 4 Operations and Algebraic Thinking (OA) Use the four operations with whole numbers to solve problems 4.OA.1
More informationMeasures of Dispersion
Lesson 7.6 Objectives Find the variance of a set of data. Calculate standard deviation for a set of data. Read data from a normal curve. Estimate the area under a curve. Variance Measures of Dispersion
More information