To determine the number of solutions for the given system of equations, we will analyze the equations using matrices and the determinant method. The equations are:
1. \( x + 2y + 3z = 1 \)
2. \( x - y + 4z = 0 \)
3. \( 2x + y + 7z = 1 \)
### Step 1: Write the system in matrix form
We can express the system of equations in the matrix form \( AX = B \), where:
\[
A = \begin{pmatrix}
1 & 2 & 3 \\
1 & -1 & 4 \\
2 & 1 & 7
\end{pmatrix}, \quad
X = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}, \quad
B = \begin{pmatrix}
1 \\
0 \\
1
\end{pmatrix}
\]
### Step 2: Calculate the determinant of matrix \( A \)
To find the number of solutions, we need to calculate the determinant of the coefficient matrix \( A \).
\[
\text{det}(A) = \begin{vmatrix}
1 & 2 & 3 \\
1 & -1 & 4 \\
2 & 1 & 7
\end{vmatrix}
\]
Using the determinant formula for a 3x3 matrix:
\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]
Where \( a = 1, b = 2, c = 3, d = 1, e = -1, f = 4, g = 2, h = 1, i = 7 \).
Calculating the determinant:
\[
\text{det}(A) = 1((-1)(7) - (4)(1)) - 2((1)(7) - (4)(2)) + 3((1)(1) - (-1)(2))
\]
\[
= 1(-7 - 4) - 2(7 - 8) + 3(1 + 2)
\]
\[
= 1(-11) - 2(-1) + 3(3)
\]
\[
= -11 + 2 + 9
\]
\[
= 0
\]
### Step 3: Analyze the determinant
Since the determinant of matrix \( A \) is 0, this indicates that the system of equations either has no solutions or infinitely many solutions.
### Step 4: Check for consistency
To determine whether the system has no solutions or infinitely many solutions, we can augment the matrix \( A \) with the matrix \( B \) and check the rank.
The augmented matrix \( [A|B] \) is:
\[
\begin{pmatrix}
1 & 2 & 3 & | & 1 \\
1 & -1 & 4 & | & 0 \\
2 & 1 & 7 & | & 1
\end{pmatrix}
\]
We can perform row operations to find the rank of this augmented matrix.
1. Subtract the first row from the second row:
\[
R_2 \rightarrow R_2 - R_1 \rightarrow (0, -3, 1 | -1)
\]
2. Subtract 2 times the first row from the third row:
\[
R_3 \rightarrow R_3 - 2R_1 \rightarrow (0, -3, 1 | -1)
\]
Now the augmented matrix looks like:
\[
\begin{pmatrix}
1 & 2 & 3 & | & 1 \\
0 & -3 & 1 & | & -1 \\
0 & -3 & 1 & | & -1
\end{pmatrix}
\]
From here, we can see that the second and third rows are identical, which indicates that the system is consistent.
### Conclusion
Since the determinant is zero and the system is consistent, the system of equations has infinitely many solutions.
