Question

The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use x, y, and z as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.

1 0 8 -1

0 1 5 2

0 0 0 0


Required:

a. What equation does the first row represent?

b. What equation does the second row represent?

c. What equation does the third row represent?

d. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.

Answer 1

a. x +8z = -1

b. y +5z = 2

c. 0 = 0

d. (x, y, z) = (-1-8z, 2-5z, z)

Explanation:

As with the original matrix, the first three columns are the coefficients of the respective variables, and the last column is the constant on the right of the equal sign.

a. The first row represents the equation ...

x + 8z = -1

__

b. The second row represents the equation ...

y + 5z = 2

__

c. The third row represents the equation ...

0 = 0

This equation expresses a truth, so indicates the original system of equations was consistent, but dependent. (a third row of 0 0 0 1 would indicate inconsistent, no solution)

__

d. The system is consistent but dependent. The solution can be written as ...

(x, y, z) = (-1-8z, 2-5z, z) . . . . where z is a free variable