Hi Mary, we hav two responses for you.
I started with the matrix
I first multiplied the first row by 1/3 to get
Now multiply the first row by -2 and add it to the second row
Can you complete it now?
An augmented matrix can be used to represent a system of equations. If we represent the two equations
as a matrix of the form
then the first column represents the x's, the second column the y's, and the last column a "constant".
Since we can add and subtract equations which are part of the same system of equations, we can do
First, subtract the second row from the first, and re-write the matrix is transformed to:
That's what's called a row operation, an operation on a row of a matrix. The result was obtained by
Now let's subtract the first row from the second row:
What does this mean? if we write the rows of the matrix back out in their full form, we'll see that
That means that y = 0.
You should now be able to complete the problem.
The idea is to use row operations on an augmented matrix, which represents a system of equations, to reduce the system to a solvable form. In general, we want the matrix to be in "reduced row-echelon form". That means that the matrix looks like
for a 2x3 matrix, where A and B are any value. That would mean that x = A and y = B. If the equation is solvable, it is possible to get the matrix into row-echelon form by means of row-operations. If you can't get it into row echelon form, either the system is unsolveable, like:
or the system has infinitely many solutions:
where A and B are not 0. Sometimes we don't have to go all the way to reduced row echelon form, as above. In your example, we only needed to perform a few row-operations to get the system into a form which could be easily dealt with, by observing that y = 0.
Using row operations on an augmented matrix to solve a system of equations is called the Gauss-Jordan method.
Hope this helps!
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