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CLASSWORK REVIEW WARM-UP t 1. Is the point (0,3) a solution for this system of inequalities? Let t = number of regular 22000 tickets 20000 x y 2 3 x 2 y 6 Let s = number of student16000 discount tickets 70t 20 s 500, 000 t s 20, 000 12000 Yes 2. Solve this linear system. 800 0 400 0 s y 3 x 4 x y 2 (1, 2) TODAY’S OBJECTIVE • To solve a system with three variables • Use the graphing calculator 3-5 SYSTEMS WITH THREE VARIABLES USING THE CALCULATOR TO SOLVE SYSTEMS • We can use amatri _______ to represent a system of equations x • Matrix -A rectangular array of numbers 3 4 2 6 1 7 A 1 3 9 4 B IDENTIFYING A MATRIX ELEMENT • Matrix ElementEach - number in a matrix • Look at the row and column numbers • Notation identifies a particular 3 element in a matrix columns a12 Is the element in row 1 and column 2 2x3 Matrix 2 rows 3 4 2 6 1 7 A 1 IDENTIFYING A MATRIX ELEMENT GRAPHING CALCULATORS Identify the indicated element. A) B) a 21 C) a13 a 22 7 1 2 3 9 1 A 2 7 6 1 4 8 7 5 Matrix of Matrix coefficient of constan s ts Number of rows = Number of equations Number of columns = 1 REPRESENT A SYSTEM USING MATRIX 2 x 7 4A 3 4 2 3 1A 1 y 5 x y X AX B X A1 B X 4 2 3 1 1 7 5 • One for the coefficients 4 x 2 y 7 3 x 1 y 5 4 x 2 y 7 3 x 1 y 5 4 2 3 1 • To use it, we need to create 2 matrices • Another for the constants REPRESENT A SYSTEM USING A MATRIX Number of rows = Number of equations Number of variables = Number of columns • We can use the graphing calculator to solve linear systems 7 5 B REPRESENT A SYSTEM USING A MATRIX 4 x 2 y 7 3 x 1 y 5 4 2 x 7 3 1 y 5 1. Write each equation with variables in the same order. Put variables on one side of the equation, constants on the other 2. Write a matrix with coefficients of the linear system. 3. Write a matrix with the constants of the linear system. REPRESENT A SYSTEM USING A MATRIX 4 x 2 y 7 3 x 1 y 5 1 x 4 2 7 y 3 1 5 Matrix of Matrix coefficient of constan s X A1 B ts 2 PRACTICE PROBLEMS 1. x 2 y 16 PRACTICE PROBLEMS 2. 7 x y 6 7 x y 6 3x y 8 1 2 1 16 3 1 8 7 1 7 1 1 6 6 Infinitely Many Solutions (0,8) SYSTEMS WITH 3 VARIABLES x y z 1 x y 3 z 3 2 x y 2 x 0 x y z 1 x y 3 z 3 2 x 0 y 4 z 4 1. Eliminate y by adding equations 1 and 2 x y 3 z 3 2x y 2z 0 3 x 0 y 5 z 3 2. Eliminate y by adding equations 2 and 3 SYSTEMS WITH 3 VARIABLES x y z 1 4 y 3 1 1 y 1 y2 5. Substitute the xvalue and z-value into one of the original equations. 3. 1x 2 y 1z 1 2 x 1z 9 3 x 1 y 3 1 2 1 1 1 2 0 1 9 3 1 0 3 (2,3,5) 4. x 4 y 6 z 21 2 x 2 y z 4 8 y z 1 4 6 1 21 1 2 2 1 0 8 1 4 1 (1,.5,3) SYSTEMS WITH 3 VARIABLES (2 x 4 z 4 ) 3 ( 3x 5 z 3 )2 6 x 12 z 12 6 x 10 z 6 2 z 6 z 3 2 x 4 z 4 2 x 4(3) 4 2 x 12 4 x4 3. Write the two new equations as a system. Eliminate x and solve for z 4. Substitute the value for z into one equation in the two variable system. Solve for x TONIGHT’S HOMEWORK Page 179; 12, 15, 18, 21, 32 - 37 The solution is (4, 2, -3) 3