Transcript 3-6 (A) Solving Systems Using Matrices

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