Write the system of equations associated with each augmented matrix . Do not solve.

Question

Write the system of equations associated with each augmented matrix . Do not solve. <4x3 Matrix>

Accepted Answer

Hey, everyone. Here we are asked to find the system of equations associated with the given augmented matrix. Here we see our augmented matrix has three rows and four columns. The first row contains the values 100 and seven. The second row has the values 010 and 15. And the third crow has the values 001 and 21. Here we have four answer choice options, answer choices A through D. Each of them being a system of three equations. One equation where X is equal to some value, then one where Y is equal to some value. And then a third equation where Z is equal to some value. So looking back at our given augmented matrix, we first need to recall that the vertical line represents an equal sign the column to the right of the vertical line represents the constant from, for each equation. And since we have three rows, we will also have three equations because each row will give us their own equation. And now from our solutions, we see that the variables we are dealing with are X Y N C. So going from alphabetical order from the leftmost column to the third column, we have the first column being the X coefficients. Then the second column are the Y coefficients and then the third column are the Z coefficients. So now with all of these components identified, we can begin writing out our three equations. Beginning with our first row, we see that the value for the X coefficient is one. So we first have one X then moving to the next column, the coefficient of Y is zero. So we have plus zero Y, then we see that the coefficient of Z is zero as well. So we have plus zero Z and then we see that we have the vertical line. So we have is equal to and the constant value to the right of the vertical line is seven. So we have one X plus zero, Y plus zero, Z is equal to seven. Now repeating this process for the last two rows, our second row gives us zero X plus one, Y plus zero, Z is equal to 15. And our third row gives us our third equation zero X plus zero, Y plus one, Z is equal to 21. And now, all that is left to do is simplify each of our equations here. We know that any coefficient of one, we do not have to include the one itself. And then we know that when we have a coefficient of zero, that the term cancels out altogether. So all of these zero terms are going to cancel. And so we are left with the three equations forming our system here where X is equal to seven Y is equal to 15 and Z is equal to 21. So with our three equations here, our three solutions for our system, we have, we are left with answer choice C where again, we have a system of three equations where X is equal to seven, Y is equal to 15 and Z is equal to 21. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Write the system of equations associated with each augmented matrix . Do not solve. <4x3 Matrix>

Verified Solution

Key Concepts

Augmented Matrix

System of Linear Equations

Row Operations