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Transcript
Thursday • Make-up Quiz #1 before next class • Submit an assignment for a late grade to clear a zero Solving Systems of Equations with Matrices Objectives • I can write matrix equations from a system of linear equations • I can solve systems of linear equations containing 3 variables using matrices • I can solve for word problems with matrices Matrix Equations • Must have all variables in the same order and all to the left of the equation sign. The constant number must be to the right of the equation sign. • ALWAYS double check this part of the problem. Systems of Equations with 3 Variables • The solution is always a triple ordered pair (x, y, z). • You may again have one solution, no solutions, or infinite solutions. Matrix Size • • • • • Every matrix is made up of: M – rows N- columns So is called a M x N matrix The following matrix is 3x3 because it has 3 rows and 3 columns 3 4 2 1 5 7 5 7 5 What Size? 2 3 1 4 2X2 2 4 5 3X1 3 4 What Size? 5 7 4 X 2 2 5 8 12 4 5 2X1 Converting Equations into Matrices • Given the following linear equations: 2x + y – z = 5 3x – 2y + z = 16 4x + 3y – 5z = 3 • 3 matrices to make the Matrix Equation: 2 1 1 3 2 1 4 3 5 Matrix A x y z 5 16 3 Matrix B Example 2: Matrix Equations • Given these equations, write a matrix equation: 3x + 4y + 2z = -9 3y – 5z = 12 2x – y = 5 3x + 4y + 2z = -9 0x + 3y – 5z = 12 2x –1y + 0z = 5 • Anytime a variable is missing, put a ZERO for its place. It’s always best to rewrite the equations with all terms before writing the matrix equation. 2 x 9 3 4 0 3 5 y 12 2 1 0 z 5 Calculator Steps • 1. Enter Matrix A data • 2. Enter Matrix B data • 3. Perform this calculation : 1 [ A] [ B] Example: Solving the Matrix Equation 2 3 1 2 Matrix A x 15 y 17 Matrix B MATRIX MODE 2nd x 1 Arrow over to EDIT With [A] selected hit ENTER Type in Matrix [A] Dimensions 2 x 2, then ENTER Enter the data for Matrix [A] GO back to MATRIX MODE 2nd x 1 Arrow over to EDIT and DOWN to Matrix [B] Hit ENTER and then Matrix [B] size 2 x 1, then ENTER Type in data for Matrix [B], then 2nd MODE (quit) Next get a blank screen 2nd MODE Calculate Solution 1 A B Now ENTER to find solution Solution is: (-3, 7) Your turn Solve by Matrices x 2 y 4 z 19 2 x y 3 z 14 3x y 2 z 5 x 2 y 4 z 19 2 x y 3 z 14 3x y 2 z 5 1 2 4 2 1 3 3 1 2 x y z 19 14 5 (1, 6, -2) Limitations • • • • • No Solution and Infinite Solutions Matrices will NOT solve You get an Error Message “Singular Matrix” You will have to look at slopes and yintercepts Word Problems • Highlight the key information • Assign variables to represent the unknown values • Write equation to reflect the data. Problem #1 • You have two jobs. One as a lifeguard and one as a cashier. Your lifeguard job pays $8 per hour and cashier pays $6 per hour. Last week you worked a total of 14 hours between the two jobs and earned $96. How many hours did you work at each job? Problem #1 Solution • You have two jobs. One as a lifeguard and one as a cashier. Your lifeguard job pays $8 per hour and cashier pays $6 per hour. Last week you worked a total of 14 hours between the two jobs and earned $96. How many hours did you work at each job? • • • • • Assign variables: L – Hours at lifeguard; C – hours at cashier Now write equations: L + C = 14 8L + 6C = 96 • Solution: (6, 8) Problem #2 • During a single calendar year, a state trooper issued 375 citations for warnings and speeding violations. There were 37 more warnings than speeding violations. How many of each citation were issued? Problem #2 Solution • During a single calendar year, a state trooper issued 375 citations for warnings and speeding violations. There were 37 more warnings than speeding violations. How many of each citation were issued? • • • • • Assign variables: W – # of warnings; S - # of speeding Now write equations: W + S = 375 W = S + 37 • Solution: (206, 169) Homework Worksheet 6-5