Section 4-6 Using Matrices to Solve Systems of Equations · Refresher on solving systems of...
Transcript of Section 4-6 Using Matrices to Solve Systems of Equations · Refresher on solving systems of...
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Warm-Up
Find the x, y and z intercepts:
a) 3𝑥 + 4𝑦 + 6𝑧 = 24
b) 2𝑥 + 5𝑦 + 10𝑧 = 10
Solve this 2-D system by Graphing on your
calculator
c) −2𝑥 + 3𝑦 = 45
4𝑥 + 5𝑦 = 10
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Solving Systems of
Equations
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Learning Targets
Refresher on solving systems of equations
Matrices
– Operations
– Uses
– Reduced Row Echelon Form
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Solving Systems of Equations
There are multiple ways to solve systems of
equations:
– Graphing
– Substitution (Equal Values Method)
– Elimination
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Solve the System by Graphing
3𝑦 − 2𝑥 = 45
5𝑦 + 4𝑥 = 10
f(x)=(2/3)x+15
f(x)=-(4/5)x+2
Series 1
-19-18-17-16-15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-1
123456789
10111213141516171819
x
y
(-8.86,9.09)
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Solve the System using Algebra
4𝑥 + 3𝑦 = 12
2𝑥 + 2𝑦 = 14
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Algebra Method cont.
Elimination Method:
4𝑥 + 3𝑦 = 12
2𝑥 + 2𝑦 = 14 4𝑥 + 3𝑦 = 12
−2(2𝑥 + 2𝑦 = 14)
4𝑥 + 3𝑦 = 12
−4𝑥 − 4𝑦 = −28
−𝑦 = −16
𝑦 = 16
2𝑥 + 2 16 = 14
2𝑥 + 32 = 14
2𝑥 = −18
𝑥 = −9
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Matrix Equations
We have solved systems
using graphing, but now
we learn how to do it
using matrices. This will
be particularly useful
when we have equations
with three variables.
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Matrix Equation
Before you start, make sure:
1. That all of your equations are in
standard form.
2. The variables are in the same
order (alphabetical usually is
best).
3. If a variable is missing use zero
for its coefficient.
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Setting up the Matrix
Equation
Given a system of equations
-2x - 6y = 0
3x + 11y = 4
Since there are 2 equations,
there will be 2 rows.
Since there are 2 variables,
there will be 2 columns.
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There are 3 parts to a matrix
equation
1)The coefficient matrix,
2)the variable matrix, and
3)the constant matrix.
Setting up the Matrix
Equation
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-2x - 6y = 0
3x + 11y = 4
The coefficients are placed
into the coefficient matrix.
2 6
3 11
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-2x - 6y = 0
3x + 11y = 4
Your variable matrix will
consist of a column.
x
y
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-2x - 6y = 0
3x + 11y = 4
The matrices are multiplied
and represent the left side
of our matrix equation.
x
y
2 6
3 11
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-2x - 6y = 0
3x + 11y = 4
The right side consists of
our constants. Two
equations = two rows.
0
4
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-2x - 6y = 0
3x + 11y = 4
Now put them together.
2 6
3 11
x
y
0
4
We’ll solve it later!
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Create a matrix equation
3x - 2y = 7
y + 4x = 8
Put them in Standard Form.
Write your equation.
3 2
4 1
x
y
7
8
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3a - 5b + 2c = 9
4a + 7b + c = 3
2a - c = 12
3 5 2
4 7 1
2 0 1
a
b
c
9
3
12
Create a matrix equation
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To solve matrix equations, get
the variable matrix alone on
one side.
Get rid of the coefficient
matrix by multiplying by its
inverse
Solving a matrix
equation
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2 6
3 11
x
y
0
4
When solving matrix equations
we will always multiply by the
inverse matrix on the left of the
coefficient and constant matrix.
(remember commutative
property does not hold!!)
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The left side of the equation
simplifies to the identity times
the variable matrix. Giving
us just the variable matrix.
x
y
2 6
3 11
10
4
2 6
3 11
12 6
3 11
x
y
2 6
3 11
10
4
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Using the calculator we can
simplify the left side. The
coefficient matrix will be A
and the constant matrix will
be B. We then find A-1B.
x
y
2 6
3 11
10
4
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The right side simplifies to give
us our answer.
x = -6
y = 2
You can check the systems by
graphing, substitution or
elimination.
x
y
6
2
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Advantages
Basically, all you have to do
is put in the coefficient
matrix as A and the constant
matrix as B. Then find A-1B.
This will always work!!!
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Solve:
Plug in the coeff. matrix as A
Put in the const. matrix as B
Calculate A-1B.
3 2
4 1
x
y
7
8
x
y
21
114
11
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Solve: r - s + 3t = -8
2s - t = 15
3r + 2t = -7
1 1 3
0 2 1
3 0 2
r
s
t
8
15
7
r
s
t
3
8
1
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Working with Matrices on TI-83, TI-84
Source: Mathbits
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Explore: • How many matrices does your calculator have?
• Use the right arrow key to move to MATH. Scroll
down and find rref. We will use this key later.
• Use the right arrow key once more to highlight EDIT.
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Step 1: Go to Matrix (above the x-1 key)
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Step 2: Arrow to the right to EDIT to allow for entering the matrix.
Press ENTER
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Step 3: Type in the dimensions (size) of your matrix and enter the elements (press ENTER).
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Step 4: Repeat this process for
a different matrix. .
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Step 5: Arrow to the right to EDIT and choose a new name.
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Step 6: Type in the dimensions (size) of your matrix and enter the elements (press ENTER).
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Using Matrices to Solve Systems of Equations:
• 1. (using the inverse coefficient matrix)
Write this system as a matrix equation and
solve: 3x + 5y = 7 and 6x - y = -8
• Step 1: Line up the x, y and
constant values.
• 3x + 5y = 7
6x - y = -8
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• Step 2: Write as equivalent
matrices.
• Step 3: Rewrite to separate out
the variables.
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Step 4: Enter the two numerical matrices in the
calculator.
Step 5: The solution is obtained by multiplying both
sides of the equation by the inverse of the matrix
which is multiplied times
the variables.
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• Step 6: Go to the home screen and enter the right
side of the previous equation.
• The answer to the system, as seen on the calculator
screen,
is x = -1 and y = 2.
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Method 2 • 2. (using Gauss-Jordan elimination method with
reduced row echelon form )
Solve this system of equations:
• 2x - 3y + z = -5
4x - y - 2z = -7
-x + 2z = -1
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• Step 1: Line up the variables and
constants
• 2x - 3y + z = -5
4x - y - 2z = -7
-x +0y + 2z = -1
• Step 2: Write as an augmented
matrix and enter into
calculator.
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• Step 3: From the home screen, choose the rref
function. [Go to
Matrix (above the x-1 key), move right→MATH,
choose B: rref]
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• Step 4: Choose name of matrix
and hit ENTER
• Step 5: The answer to the system, will be the last
column on the calculator screen:
x = -3
y = -1 z = -2.
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Method 2: • Case 1: Unique solution
−2𝑟 + 2𝑠 + 5𝑡 = −3
−𝑟 + 5𝑠 + 4𝑡 = −15
−𝑟 + 3𝑠 + 𝑡 = −6
Enter as a
3X4 matrix
𝑟 = −4
𝑠 = −3
𝑡 = −1
Diagonal is all ones so
there is a solution:
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Method 2: • Case 2: No solution
𝑥 + 5𝑦 − 𝑧 = 21
−3𝑥 + 𝑦 − 3𝑧 = −28 5𝑥 + 𝑦 + 4𝑧 = 3
Enter as a
3X4 matrix
Last row:
0 0 0 1
No solution.
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Method 2: • Case 3: Infinitely Many Solutions
−5𝑚 + 𝑛 − 2𝑝 = −22 𝑚 + 3𝑛 − 6𝑝 = 14 −6𝑚 + 2𝑛 − 4𝑝 = −24
Enter as a
3X4 matrix
Last row:
0 0 0 0 Infinitely Many Solutions
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Website to Visualize the Solutions
• http://www.cpm.org/flash/technology/3dsystems.s
wf
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For Tonight
• Intro to Matrices Worksheet