Context: In the equation $3x + 21 = 3x + k$, $k$ is a constant, and the equation is said to have infinitely many solutions. We need the value of $k$ that makes this true.

Answer: 21

Why: A linear equation has infinitely many solutions exactly when the two sides are identical, so that every value of $x$ works. Subtract $3x$ from both sides: the variable terms cancel, leaving $21 = k$. So the equation is an identity only when $k = 21$.

Verification: With $k = 21$ the equation reads $3x + 21 = 3x + 21$, which is true for every value of $x$, so there are infinitely many solutions. For any other value of $k$, subtracting $3x$ leaves the false statement $21 = k$, and the equation would have no solutions at all.

Common slips: Trying to solve for $x$ (the $x$-terms cancel, so there is no single solution to find), or mixing up the conditions: matching constants gives infinitely many solutions, while unequal constants with equal $x$-coefficients gives no solution.