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Solve a system of linear equations by representing the system with a matrix. 3-6 (A) SOLVING SYSTEMS USING MATRICES Matrix Rectangular array of numbers Usually displayed within brackets Dimensions are the number of rows by columns in the array. Each number in a matrix is called a matrix element. Elements can be identified by the row and column. Ex: 𝑎12 is the element in row 1 column 2 which is 4 Systems of Linear Equations You can solve a system by using matrix. Each matrix row represents an equation Use coefficients, leaving out variables and replace the equal sign with a vertical bar. Ex: x + 3y =7 and 3x + y = -8 1 3 7 3 1 −8 Column 1 shows x-coefficients, column 2 ycoefficients, and column 3 constants. Writing a System From a Matrix Write the system of linear equations shown by the following matrix. 5 2 7 0 1 9 5x + 2y = 7 0x + 1y = 9 5𝑥 + 2𝑦 = 7 𝑦=9 Row Operations Your goal is to get a matrix in the form 1 0 𝑎 0 1 𝑏 To get here you can: Switch any 2 rows Multiply an entire row by a constant Add one row to another Combine any of these steps Solving a System with a Matrix 𝑥 + 4𝑦 = −1 2𝑥 + 5𝑦 = 4 1 4 −1 2 5 4 Multiply row 1 by -2 and add row 1 to row 2 1 4 −1 0 −3 6 1 1 4 −1 Multiply row 2 by − so 3 0 1 −2 Multiply row 2 by -4 and add row 2 to row 1 1 0 7 so the solution is (7, -2) 0 1 −2 Write in matrix form: You Try! 9𝑥 − 2𝑦 = 5 3𝑥 + 7𝑦 = 17 (1,2) 3-6 (B) THREE VARIABLES Systems In 3 Variables Use a 3 x 4 matrix Column 1 is x-coefficients Column 2 is y-coefficients Column 3 is z-coefficients Same rules apply Goal is to get a matrix of the form 1 0 0 1 0 0 0 𝑎 0 𝑏 1 𝑐 Practice 2𝑎 + 3𝑏 − 𝑐 = 1 −4𝑎 + 9𝑏 + 2𝑐 = 8 −2𝑎 + 2𝑐 = 3 2 3 −1 1 Write as a matrix: −4 9 2 8 −2 0 2 3 −𝟏 Go to 2nd 𝒙 (MATRX) then move right to EDIT Select a matrix and enter the dimensions (3 x 4) Now fill in the matrix 2nd 𝒙−𝟏 (MATRX) then move right to MATH Scroll down to B:rref( then ENTER once Now 2nd 𝒙−𝟏 (MATRX) then select the name of the matrix you used (probably the first one [A]) then ENTER twice. If you get decimals, go to MATH then 1:FRAC so the answers will now be in fraction form. You Try! 𝑥 + 4𝑦 + 6𝑧 = 21 2𝑥 − 2𝑦 + 𝑧 = 4 −8𝑦 + 𝑧 = −1 (1, ½ , 3) Assignment Odds p.179 #9-23, 27, 29, 33-37, 40