The points \( \mathrm{A}(5,4) \) and \( \mathrm{B}(9,-2) \) lie on the blue line, labelled F , shown in the diagram above. Find an equation of the line \( F \).

To find the equation of the line \( F \) that passes through the points \( \mathrm{A}(5,4) \) and \( \mathrm{B}(9,-2) \), we first need to calculate the slope \( m \) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in our points: \[ m = \frac{-2 - 4}{9 - 5} = \frac{-6}{4} = -\frac{3}{2} \] Now we can use the point-slope form of a line equation, which is \( y - y_1 = m(x - x_1) \). We can use point \( \mathrm{A}(5,4) \): \[ y - 4 = -\frac{3}{2}(x - 5) \] Distributing the slope: \[ y - 4 = -\frac{3}{2}x + \frac{15}{2} \] Adding 4 to both sides: \[ y = -\frac{3}{2}x + \frac{15}{2} + 4 \] To express 4 as a fraction with a denominator of 2, we convert it: \[ 4 = \frac{8}{2} \] Thus, our equation becomes: \[ y = -\frac{3}{2}x + \frac{15}{2} + \frac{8}{2} = -\frac{3}{2}x + \frac{23}{2} \] So, the equation of the line \( F \) is: \[ y = -\frac{3}{2}x + \frac{23}{2} \]