Consider the following system of equations: $$\begin{aligned} 3x + 2y &= 1 \\ 4x + 7z &= 1 x + y + z &= 3 x - 2y + 7z &= 0 \end{aligned}$$ The number of solutions for this system is ________

Consider the 4 equations in 3 variables $(x, y, z)$:

1. $3x + 2y = 1$

2. $4x + 7z = 1$

3. $x + y + z = 3$

4. $x - 2y + 7z = 0$

Notice that subtracting equation (1) from equation (2) gives:

This is identically equation (4), so equation (4) is linearly dependent on equations (1) and (2).

Now solve the first three equations: From (1): $y = \frac{1 - 3x}{2}$ From (2): $z = \frac{1 - 4x}{7}$

Substitute $y$ and $z$ into equation (3):

Multiply the entire equation by 14:

This gives unique values:

Since this unique solution satisfies all 4 equations, the system is consistent and has exactly 1 unique solution.