Question

Use Row Operations to solve the system

X + Y- Z = 7

2X - Y + Z= -10

X - 4Y + 3z = -43

Select the correct choice and fill in any answer boxes within your choice.

A. There is one solution. The solution is (_____, _____, _____)

exact answer in simplified form

B. There are infinitely many solutions. The solutions are (_____, _____,Z),

where z is any real number.

C. There is no solution.

Answer 1

Using row operations on the system of equations leads to a single unique solution, which is (-25, -2, 30). Therefore, the correct option is A).

Step-by-step explanation:

To solve the system of equations using row operations, we start by setting up our augmented matrix:

[ 1 1 -1 | 7]
[ 2 -1 1 | -10]
[ 1 -4 3 | -43]

We want to create zeros below the leading 1 in the first column of the first row, to start forming an upper triangular matrix. This can be achieved by performing the following row operations:

1. R2 = R2 - 2*R1
2. R3 = R3 - R1

After performing these operations, our matrix should look like this:

[ 1 1 -1 | 7]
[ 0 -3 3 | -24]
[ 0 -5 4 | -50]

Next, we create a zero below the leading term in the second column of the second row:

1. R3 = R3 + (5/3)*R2

Our matrix now:

[ 1 1 -1 | 7]
[ 0 -3 3 | -24]
[ 0 0 -1 | -30]

From here, we can back-substitute to solve for Z, then Y, and finally X. Doing so yields the following solutions:

• Z = 30
• Y = -2
• X = -25

Therefore, there is one solution to the system and the solution is (-25, -2, 30).

We can mention the correct option in the final part of our solution process, which is:

A. There is one solution. The solution is (-25, -2, 30)