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Using Matrices to
Solve Systems of
Equations
Matrix Equations
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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.
Matrix Equation
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Before you start, make
sure that both of your
equations are in standard
form and the variables are
in the same order
(alphabetical usually is
best).
Setting up the Matrix
Equation
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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.
Setting up the Matrix
Equation
There are 3 parts to a matrix
equation
1)The coefficient matrix,
2)the variable matrix, and
3)the constant matrix.
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-2x - 6y = 0
3x + 11y = 4
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The coefficients are placed
into the coefficient matrix.
2

 3
6

11
-2x - 6y = 0
3x + 11y = 4
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Your variable matrix will
consist of a column.
x 
 
y 
-2x - 6y = 0
3x + 11y = 4
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The matrices are multiplied
and represent the left side
of our matrix equation.
2 6 x 

  
 3 11 y 
-2x - 6y = 0
3x + 11y = 4
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The right side consists of
our constants. Two
equations = two rows.
0 
 
4 
-2x - 6y = 0
3x + 11y = 4
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Now put them together.
2 6x  0 

    
 3 11y  4
We’ll solve it later!
Create a matrix equation
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3x - 2y = 7
y + 4x = 8
Put them in Standard Form.
Write your equation.
3 2 x  7

   
4 1 y  8
Create a matrix equation
3a - 5b + 2c = 9
4a + 7b + c = 3
2a - c = 12
3 5 2 a 9 

   
4 7 1 b  3 

   
2 0 1c  12 
Solving 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
2

 3
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6x  0 
    
11y  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!!)
2 6 2 6x  2 6 0

 
  
  
 3 11  3 11 y   3 11  4
1
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1
The left side of the equation
simplifies to the identity times
the variable matrix. Giving
us just the variable matrix.
x  2 6 0
   
  
y   3 11 4
1
x  2 6 0
   
  
y   3 11 4
1
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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  6
   
y  2 
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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.
Advantages
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Basically, all you have to do
is put in the coefficient
matrix as A and the constant
-1
matrix as B. Then find A B.
This will always work!!!
NO SOLVING FOR Y!!!!! :)
3 2 x  7







Solve: 4 1 y  8
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Plug in the coeff. matrix as A
Put in the const. matrix as B
Calculate A-1B.
 1 
x  2 
11
   
y  4 

11 

r  3
l r - s + 3t = -8
   
s   8 
l 2s - t = 15
   
l 3r + 2t = -7
t   1 
1 1 3 r 8

   
0 2 1s  15

   
3 0 2 t  7
Solve:
Word Problem Systems
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The sum of three numbers
is 12. The 1st is 5 times
the 2nd. The sum of the
1st and 3rd is 9. Find the
numbers.
Word Problem Systems
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The sum of three numbers is
12.
x + y + z = 12
The 1st is 5 times the 2nd.
x = 5y
The sum of the 1st and 3rd is 9.
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x+z=9
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Word Problem Systems
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x + y + z = 12
x = 5y => x - 5y = 0
x+z=9
1 1 1 x  12

   
1 5 0 y  0 

   
1 0 1 z 9 
Word Problem Systems
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x + y + z = 12
x - 5y = 0
x+z=9
x  15 
   
y  3 
   
z  6