Writing Systems of Equations as
Augmented Matricies
What is an augmented matrix?
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An augmented matrix is essentially two matrices put
together.
In the case of a system of linear equations, it’s composed
of the coefficients of the equations.
Essentially, it’s just the system of equations without the
variables or the plus, minus, or equals signs.
What’s the point?
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Matrices are a compact way of presenting information
Matrices are easy to work with using elementary row
operations
Steps
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Put your variables on one side, constants on the other.
Find your coefficients.
Form a matrix using the coefficients
Augment the matrix by adding the constant terms to
the right side.
Step 1: Put variables on one side
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In general, you want your equations to be in the form ax
+ by = c or ax + by + cz = d, where a, b, c, and d are
constants and x, y, and z are your variables.
Make sure all your equations have the variables in the
same order!
Example Step 1:
Say you have the system of equations
3x = 7 + 5y
2y = 3x + 2
Put this in a form useful for converting these equations to
an augmented matrix.
In the first equation, subtract 5y from both sides. In the
second equation, subtract 3x from both sides.
3x - 5y = 7
-3x + 2y = 2
Note that the two variable terms are in the same order
in both cases.
Step 2: Find The Coefficients
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The coefficients are the constants that are multiplied by
your variables.
Keep any negative signs.
If a variable is not present in a particular equation, its
coefficient is zero.
Example Step 2
Recall our system of equations:
3x - 5y = 7
-3x + 2y = 2
The coefficients of the first equation are 3 and -5. The
coefficients of the second equation are -3 and 2.
Step 3: Put the coefficients in a matrix
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Each equation forms a row of the matrix .
Each variable forms a column.
Sample 3x3 matrix (not based on our system of
equations):
Example Step 3
Remember, the coefficients of our first equation are 3 and
-5. The coefficients of our second equation are -3 and 2.
Our matrix, then, is:
Step 4: Augment the Matrix
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Draw a dotted line down the right side of your matrix.
Add the constant term of each equation to the right of
the line.
The result should look like this:
Example Step 4
Recall that our matrix is
Our two constant terms are 7 and 2 in equation 1 and
equation 2, respectively. Thus, our augmented matrix is