3Linear Systems and Matrices 3.1 Solve Linear Systems by Graphing 1
Find the solution of the systems by looking at the graphs 2
Decide whether the ordered pair is a solution of the system of linear equations: 1. 5x + y = 19 x 7y = 3 ( 2,1) 2. x + 3y = 15 4x + y = 6 (3, 6) 3
Solving System of Equations by Graphing 4
Example 1: ex: x + y = 3 y = 2x 6 5
Example 2: y = 2x 4 y = ½x + 1 6
Example 3: y = 2x + 2 4x y = 6 7
Special Type of Linear Systems A system that has at least one solution is consistent. If the system has exactly one solution it is consistent and independent, while a system that has infinitely many solutions is dependent A system that has no solution is inconsistent. Lines Intersect at 1 pt ONE SOLUTION Independent Consistent Lines are parallel (same slope) never intersect! NO SOLUTION Inconsistent Same Line Infinitely many solutions Dependent Consistent 8
Example 1: 2x + y = 5 6x 3y = 15 9
Example 2: 6x + 2y = 4 9x + 3y = 12 10
Example 3: 2x + y = 7 3x y = 2 11
You Try! 1. x + y = 7 2x 2y = 18 2. 4x + y = 8 12x + 3y = 24 3. 4x + y = 8 2x 2y = 14 12
3.2 Solve Linear Systems by Algebraically Solving Systems Using Substitution 3x + y = 7 x = 2 5x 2y = 8 y = 6 13
Basic Examples: y = x 4 4x + y = 26 y = x + 4 2x + y = 19 14
More Challenging Examples: 15
Solving Using Elimination: x + y = 1 x y = 2 x+y = 2 x y = 0 16
More Challenging Examples: 17
Special Types of Solutions: 18
Word Problems: Ms. Hamilton tells you that next weeks test is worth 100 points and contains 38 problems. Each problem is worth either 5 points or 2 points. Because you are studying system of linear equations your teacher says that for extra credit you can figure out how many problems of each value are on the test. How many of each value are there? 19
Word Problems: Last year you mowed grass and shoveled snow for 10 households. You earned $200 per household mowing for the entire season and $180 per household shoveling the entire season. If you earned a total of $1880 last year, how many households did you mow and shovel for? 20
Word Problems: A travel agency offers two Boston outing plans. Plan A includes hotel accommodations for three nights and two pairs of Red Sox Tickets worth $518. Plan B includes hotel accommodations for five nights and four pairs of Red Sox Tickets worth $907. Write a system of equations that determines how much one night at a hotel and one pair of tickets cost. 21
3.3 Graph Systems of Linear Inequalities System of linear inequalities A solution of a system of inequalities is an ordered pair that is a solution of each inequality in the system 22
Example of system of linear inequalities with NO SOLUTION 23
Graph a system with an abolute value inequality 24
Example 1: 25
Example 2: 26
Example 3: 27
Example 4: 28
3.4 Solve System of Linear Equations in Three Variables A linear equation in three variables x, y, and z is an equation of the form ax +by + cz = d where a, b, and c are all not zero. The following is an example of a system of three linear equations in three variables. equation 1 equation 2 equation 3 A solution of such a system is an ordered triple (x, y, z) whose coordinates make each equation true. 29
Example 1: Use elimination method 30
Example 2: Solve a three variable system with no solution 31
Example 3: Solve a three variable system with no solution 32
3.5 Perform Basic Matrix Operations A matrix is a rectangular arrangement of numbers in rows and columns. The dimensions of a matrix with m rows and n columns are m x n (read: m by n). The numbers in a matrix are it's elements. A = the element in the first row and third column is 5 } 2 rows } 3 columns 33
Adding Matrices Key Concept: Adding and Subtracting Matrices Subtracting Matrices Key Concept: Properties of Matrix Operations Let A, B and C be matrices with the same dimensions, and let k be a scalar (a real number) Associative Property of Addition (A + B) + C = A + (B + C) Commutative Property of Addition Distributive Property of Addition Distributive Property of Subtraction A + B = B + A k(a + B) = ka + kb k(a B) = ka kb 34
Example 1: 35
Example 2: 36
Example 3: 37
Example 4: 38
Solve the matrix equation for x and y: 39
3.6 Multiply Matrices The product of two matrices A and B is defined provided the number of columns in A is equal to the number of rows in B. If A is an m x n matrix and B is an n x p matrix. Then the product of AB is an m x p matrix m x n n x p mx p equal dimensions of AB Examle: Describe matrix products. a) A: 4 x 3, B: 3 x 2 b) A: 3 x 4; B: 3 x 2 40
Key Concept: Multiplying Matrices Words To find the element of the ith row and jth column of the product matrix AB, multiply each element in the ith row of A by the corresponding element in the jth column of B, then add the products Algebra Properties of Matrix Multiplication let A, B, and C be matrices and let k be a scalar Associative Property: Left Distributive Property: Right Distributive Property: Associative Property of Scalar Multiplication: 41
Example 1: Find the product of two matrices 42
Example 2: Use matrix operations 1. 3AB 2. AB + AC 3. AB BA 43
Example 3: Matrix Multiplication in the Calculator 44
3.7 Evaluate Determinants Key Concept: The Determinant of a Matrix Determinant of a 2 x 2 Matrix Determinant of a 3 x 3 Matrix The determinant of a 2 x 2 matrix is the difference of the products of the elements on the diagonals. 3 x 3: Repeat first two columns to the rightof the determinant then subtract the red products from the sum of the blue products 45
Example 1: Find determinant of 2 x 2 matrix 46
Example 2: Find determinant of 3 x 3 matrix 47
Example 3: Area of a Triangle You can use a determinant to find the area of a triangle whose vertices are points in a coordinate plane The area of a triangle with vertices (x 1, y 1 ), (x 2, y 2 ), and (x 3, y 3 ) is given by: example: Find the area of the triangle with vertices A(5,11) B(9,2) and C(1,3) 48
3.8 Use Inverse Matrices and Solve Linear Systems The n x n identity matrix is a matrix with 1's on the main diagonal and 0's elsewhere. If A is any n x n matrix and I is the n x n Identity matrix. Then AI = A and IA = A 2 X 2 Identity Matrix 3 X 3 Identity Matrix Two n xn matrices A and B are inverses of each other if their product (in both orders) is the n x n identity matrix. That is AB = I and BA = I. An n x n matrix A has an inverse if det A 0. The symbol for the inverse of A is A 1 49
Key Concept: The Inverse of a 2 x 2 Matrix The inverse of the matrix is. Provided ad cb 0 50
Example 1: Find the inverse of 2 x 2 matrix 51
Example 2: Solve a Matrix equation AX = B 52
Example 3: Use a calculator to find the inverse of a 3 x 3 matrix 53
Key Concept: Using an inverse matrix to solve a linear system Step 1: Write the system as a matrix equation AX = B. The matrix A is the coefficient matrix. X is the matrix of variables, and B is the matrix of constants. Step 2: Find the inverse of matrix A. Step 3: Multiply each side of AX = B by A 1 on the left to find the solution X = A 1 B 54
Example 4: Solve a linear system 55
Gifts: A company sells three types of movie gift baskets. A basic basket with 2 movie passes and 2 package of microwave popcorn costs $15.50. A medium basket with 2 movie passes, 2 packages of popcorn, and 1 DVD costs $37. A super basket with 4 movie passes, 3 packages of popcorn, and 2 DVDs costs $72.50. Find the cost of each item in the gift basket. 56
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