The system kx + 4y = 20 and 3x + 2y = 10 has infinitely many solutions. What is the value of k?

For a system to have infinitely many solutions, the two equations must describe the same line, so one is a constant multiple of the other. Compare $3x + 2y = 10$ with $kx + 4y = 20$. The $y$-coefficient doubles from 2 to 4 and the constant doubles from 10 to 20, so the multiplier is 2. The $x$-coefficient must double as well: $k = 2 \times 3 = 6$. Check: $6x + 4y = 20$ is exactly twice $3x + 2y = 10$. A common wrong answer is 3, which copies the original coefficient without scaling. Another is 10, the constant.