In the \( x y \)-coordinate system, points \( (2,9) \) and \( (-1,0) \) lie on the straight line \( k \). If the point \( (n, 21) \) lies on line \( k \), what is the value of \( n \) ?

To find the value of \( n \) where the point \( (n, 21) \) lies on the line passing through the points \( (2, 9) \) and \( (-1, 0) \), we first need to determine the equation of the line. 1. **Calculate the slope \( m \) of the line** using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in the points \( (2, 9) \) and \( (-1, 0) \): \[ m = \frac{0 - 9}{-1 - 2} = \frac{-9}{-3} = 3 \] 2. **Use the point-slope form of the equation of the line** \( y - y_1 = m(x - x_1) \). We can use the point \( (2, 9) \): \[ y - 9 = 3(x - 2) \] Expanding this gives: \[ y - 9 = 3x - 6 \] So, rearranging the equation results in: \[ y = 3x + 3 \] 3. **Substitute \( y = 21 \)** into the line's equation to find \( n \): \[ 21 = 3n + 3 \] Subtract 3 from both sides: \[ 18 = 3n \] Divide by 3: \[ n = 6 \] Therefore, the value of \( n \) is \( \boxed{6} \).