Transcript Section 3.6 – Solving Systems Using Matrices

Section 3.6 –
Solving Systems Using Matrices
Students will be able to:
•Represent a system of linear equations with a
matrix
•To solve a system of linear equations using
matrices
NEW VOCABULARY:
Matrix
Matrix Element
Row Operations
Section 3.6 –
Solving Systems Using Matrices
Section 3.6 –
Solving Systems Using Matrices
An array of numbers, such as each of those
suggested by the tile arrangements in the
Solve It, is a matrix.
Essential Understanding
You can use a matrix to represent and solve a
system of equations without writing ANY
variables.
REAL LIFE
Section 3.6 –
Solving Systems Using Matrices
A matrix is a rectangular array of numbers. You usually
display the array with brackets. The dimensions of a
matrix are the number of rows and columns in the array.
Each number is a matrix is a matrix element. You can
identify a matrix element by its row and column of
numbers. In Matrix A, a12 is the element in row 1 and
column 2. a12 is the element 4.
Section 3.6 –
Solving Systems Using Matrices
You can represent a system of equations
efficiently with a matrix.
Each matrix row represents an equation.
The last matrix column shows the constants to
the right of the equal signs.
Each of the other columns shows the coefficients
of one of the variables.
Section 3.6 –
Solving Systems Using Matrices
Section 3.6 –
Solving Systems Using Matrices
Problem 1:
How can you represent the system of equations
with a matrix?
2x + y = 9
x – 6y = -1
Section 3.6 –
Solving Systems Using Matrices
Problem 1:
How can you represent the system of equations
with a matrix?
x – 3y + z = 6
x + 3z = 12
y = -5x + 1
Section 3.6 –
Solving Systems Using Matrices
Problem 1:
How can you represent the system of equations
with a matrix?
-4x – 2y = 7
3x + y = -5
Section 3.6 –
Solving Systems Using Matrices
Problem 1:
How can you represent the system of equations
with a matrix?
4x – y + 2z = 1
y + 5z = 20
2x = -y + 7
Section 3.6 –
Solving Systems Using Matrices
Problem 2:
What linear system of equations does this matrix
represent?
5 2 7


0
1
9


Section 3.6 –
Solving Systems Using Matrices
Problem 2:
What linear system of equations does this matrix
represent?
 2 0 6


5

2
1


Section 3.6 –
Solving Systems Using Matrices
You can use a matrix that represents a system of
equations to solve the system. In this way, you
do not have to write the variables.
To solve the system using a matrix, use the steps
for solving by elimination.
Each step is a row operation.
Section 3.6 –
Solving Systems Using Matrices
Your goal is to use row operations to get a matrix
in the form:
Notice that the first matrix represents the system
x = a, y = b, which then will be the solution of a
system of two equations in two unknowns. The
second matrix represents the system x = a, y = b,
and z = c.
Section 3.6 –
Solving Systems Using Matrices
Section 3.6 –
Solving Systems Using Matrices
Problem 3:
What is the solution of the system?
x + 4y = -1
2x + 5y = 4
Section 3.6 –
Solving Systems Using Matrices
Problem 3:
What is the solution of the system?
x+y=5
-2x + 4y = 8
Section 3.6 –
Solving Systems Using Matrices
Problem 3:
What is the solution of the system?
x +3y = 22
2x - y = 2
Section 3.6 –
Solving Systems Using Matrices
Matrices that represent the solution of a system
are in reduced row echelon form. Many
calculators have a rref(reduced row echelon
form) function for working with matrices.
This function will do all the row operations for
you. You can use the rref to solve a system of
equations.
Section 3.6 –
Solving Systems Using Matrices
What is the solution of the system of equations?
2a + 3b – c = 1
-4a + 9b + 2c = 8
-2a + 2c = 3
Section 3.6 –
Solving Systems Using Matrices
What is the solution of the system of equations?
a + 4b + 6c = 21
2a – 2b + c = 4
-8b + c = -1
Section 3.6 –
Solving Systems Using Matrices