The system of equations 2x + 5y = 10 and 6x + ky = 30 has infinitely many solutions. What is the value of k?

A system has infinitely many solutions when the two equations describe the same line, so the second must be a constant multiple of the first. Comparing the $x$-coefficients, $6 = 3 \cdot 2$, and comparing the constants, $30 = 3 \cdot 10$, so the multiplier is 3. The $y$-coefficient must then be $3 \cdot 5 = 15$, so $k = 15$, choice C. Choice A (5) copies the original $y$-coefficient without scaling. Choice B (10) copies the constant. Choice D (18) results from multiplying by the wrong factor, such as $6 \cdot 3$. Any value other than 15 gives either one solution or none.