\( \begin{array}{l} k \text { : }\\ \left\{\begin{array}{l} x+2 k y+5 k z=1 \\ x+4 y+10 z=1 \end{array}\right. \end{array} \)

To solve the given system of equations for \( k \), we can start by simplifying both equations. The first equation is dependent on \( k \), while the second one is constant. By eliminating \( x \) from both equations, we can compare the terms related to \( k \). Since the second equation is a homogeneous form of the first, they should be equal when expressed in terms of \( k \). This leads to establishing a relationship involving \( k \) by manipulating the coefficients. Setting these equal can help us isolate \( k \) and find potential solutions. Now, remember that \( y \) and \( z \) can be treated as variables that are independent of \( k \), so make sure to account for all variables involved carefully when extracting solutions. By examining the relationship between these two equations, we can analytically derive the values for \( k \) if we proceed and compare coefficients properly.