System Of Linear Equations

Step 1: Write the system of equations clearly: We will solve this system step by step.

Step 2: From equation (1), solve for : Substitute equation (4) into equations (2) and (3) to simplify the system. \bigskip Step 3: Substituting into equation (2): Now substitute equation (5) into equation (3).

Step 4: Substituting into equation (3): Now subtract equation (5) from equation (6):

Step 5: Since we encounter a contradiction, the system of equations has no solution.

Step 1: Construct the Coefficient Matrix

The coefficient matrix is:

For a non-zero solution to exist, the determinant of must be zero:

Step 2: Compute the Determinant

Expanding along the first column:

Computing Each Minor Determinant

Substituting:

Step 3: Conclusion

For a non-trivial solution to exist, we require:

Thus, the correct answer is:

Step-by-Step Determinant Calculation

Step 1: Set up the determinant of the coefficient matrix

For the homogeneous system to have a non-trivial solution, the determinant of the coefficient matrix must be zero. The coefficient matrix is:

Step 2: Compute the determinant

Using the formula for the determinant of a matrix, we calculate:

Expanding each of the determinants:

Step 3: Set the determinant to zero

Setting the determinant expression to zero for non-trivial solutions:

Step 4: Relate to given expression

Plugging the expression for into :

To satisfy the equation given in the options, set this equal to 6:

Substituting back into the expression for :

Now, calculate with these values:

Thus, , confirming option (2).

Step 1: Solve the system in standard form

Rearrange the equations as follows:

3x - 4y + 2z = -7

2x + 3y - z = 10

x - 2y - 3z = 3

You can apply any standard method (such as substitution, elimination, or using matrices) to solve for the values of (x, y, z).

Step 2: Verify the x-value α

After solving the system, we determine that:

α = x = 1