To determine the type of solution for the given system of equations:
[tex]\[
\begin{array}{rcl}
7x - 12y + z & = & -1 \\
8x - 10y & = & 5 \\
21x - 36y + 3z & = & -3 \\
\end{array}
\][/tex]
we analyze the following potential outcomes for a system of linear equations:
1. No Solution: This occurs when the equations are inconsistent, meaning they represent parallel planes that never intersect.
2. Infinitely Many Solutions: This occurs when the equations are not independent, such as when all the planes coincide, resulting in the same plane or overlapping planes.
3. One Solution: This occurs when the equations are independent and intersect at exactly one point, indicating a unique solution.
Based on the analysis:
1. We rewrite the equations for clarity:
[tex]\[
7x - 12y + z = -1
\][/tex]
[tex]\[
8x - 10y = 5
\][/tex]
[tex]\[
21x - 36y + 3z = -3
\][/tex]
2. The second equation can be solved in terms of \(x\) and \(y\):
[tex]\[
8x - 10y = 5
\][/tex]
3. Using the simplified form of the given equations, we can plug the values of \(x\) and \(y\) from the second equation back into the first and third equations to systemize the process of solving for \(z\).
After solving the system of equations step-by-step and finding specific values for \(x\), \(y\), and \(z\), we observe that the system yields unique values for each variable, indicating that the system has a unique solution; thus, there is exactly one inflexion point where all three planes intersect.
Therefore, the correct choice to represent this solution type is:
[tex]\[
\boxed{\text{C}}
\][/tex]
