Questions: Solve the system of linear equations using matrices. x+y = 1 6y = 12 7x+3y-2z = 5 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The solution is ( . . ). (Type an exact answer in simplified form.) B. There are infinitely many solutions. C. There is no solution.

Solve the system of linear equations using matrices. x+y = 1 6y = 12 7x+3y-2z = 5 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The solution is ( . . ). (Type an exact answer in simplified form.) B. There are infinitely many solutions. C. There is no solution.
Transcript text: Solve the system of linear equations using matrices. \[ \left\{\begin{array}{rrr} x+y & = & 1 \\ 6 y & = & 12 \\ 7 x+3 y-2 z & = & 5 \end{array}\right. \] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The solution is ( $\square$ . $\square$ $\square$ ). (Type an exact answer in simplified form.) B. There are infinitely many solutions. C. There is no solution.
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Solution

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Solution Steps

Step 1: Matrix Representation

The augmented matrix is represented as:

[[ 1 1 0 1] [ 0 6 0 12] [ 7 3 -2 5]]

Step 2: Row Reduction

After applying row operations, we get a matrix in reduced row echelon form.

Step 3: Solution Analysis

The system has exactly one solution since the rank of the coefficient matrix equals the rank of the augmented matrix and the number of variables.

Final Answer:

The solution is:

[-1. 2. -3.]

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