Final answer:
To solve the system of linear equations, we can use Gaussian elimination method. By performing row operations on the augmented matrix, we can transform it into an upper triangular matrix. The system of equations has infinitely many solutions as the last row of the matrix represents the equation 0 = 0.
Step-by-step explanation:
To solve the system of linear equations using Gaussian elimination method, we can write the augmented matrix and perform row operations to transform it into an upper triangular matrix. Let's represent the given system of equations as:
4x + 3y + 6z = 25
Next, we can write the augmented matrix as:
[4 3 6 | 25]
To convert the augmented matrix into an upper triangular matrix, we can perform row operations. The goal is to eliminate the coefficients below the diagonal. Let's begin with the first row:
[4 3 6 | 25] (Step 1: Divide Row 1 by 4)
[1 3/4 6/4 | 25/4]
Next, we can eliminate the coefficient below the diagonal:
[1 3/4 6/4 | 25/4] (Step 2: Multiply Row 1 by -3/4 and add to Row 2)
[1 3/4 6/4 | 25/4]
[0 0 3/4 | 25/4]
Finally, we can continue eliminating the coefficients below the diagonal:
[1 3/4 6/4 | 25/4] (Step 3: Multiply Row 1 by -6/4 and add to Row 3)
[1 3/4 6/4 | 25/4]
[0 0 0 | 0]
The system of equations is now in upper triangular form. We can back-substitute to find the values of x, y, and z. Since the last row represents the equation 0 = 0, this equation is satisfied for all values of x, y, and z. Therefore, the system of equations has infinitely many solutions.
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