Linear Systems and Augmented
Matrices
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 and their constant
terms.
Essentially, it’s just the system of equations without the
variables or the plus, minus, or equals signs.
Translating a System to a Matrix
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In each equation, put your variable terms 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, separated by a dotted line.
Why Matrices?
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Matrices are a compact way of presenting information.
Matrices are easy to work with using elementary row
operations.
Example
Let’s go through the process of converting a system of
linear equations to an augmented matrix for this system
of equations:
2x = 3 – 5y
-7x + y = 0
Step 1: Organize Your Equations
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In general, when you have two or three variables, you
want your equations to be in the form ax + by = c or ax
+ by + cz = d, respectively. 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!
In our system, all we have to do is add 5y to both sides of
the first equation, leaving
2x + 5y = 3
-7x + y = 0
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.
In our system of equations,
2x + 5y = 3
-7x + y = 0
Our coefficients are 2 and 5 in the first equation, -7 and 1
in the second.
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.
For our system of linear equations,
2x + 5y = 3
-7x + y = 0
The matrix will be
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.
In our system of equations, the constants in the first and
second equations are 3 and 0, respectively. Thus, the final
augmented matrix will look like this: