Determine whether the system of equations is in row-echelon form.
x-y+3z=-11
y+8z=-12
z=-2
To determine whether the system of equations is in row-echelon form, we must check if it satisfies the following conditions:
1. Leading Entries: Each row should have a leading entry (the leftmost non-zero entry) of 1. The leading entry must be to the right of the leading entry in the row above it.
2. Zero Rows: Any row consisting entirely of zeros should be at the bottom of the system.
3. Non-zero Rows: All non-zero rows should be above any zero rows.
4. Column Zeroes: Any column containing a leading entry must have zeroes in all other positions.
Now let's analyze the given system:
x - y + 3z = -11
y + 8z = -12
z = -2
We can observe the following:
1. Leading Entries: The leading entries in each row are 1 for the first row (in the x column), 1 for the second row (in the y column), and 1 for the third row (in the z column). So, this condition is satisfied.
2. Zero Rows: There are no rows consisting entirely of zeros. So, this condition is satisfied.
3. Non-zero Rows: All rows are non-zero. So, this condition is satisfied.
4. Column Zeroes: In the x column, there are no other entries aside from the leading entry. In the y column, there is no entry to the left of the leading entry. In the z column, there are no other entries aside from the leading entry. So, this condition is satisfied.
Based on the analysis above, we can conclude that the given system of equations is in row-echelon form.