A line passes through the points \( (5,-7.5) \) and has a \( y \)-intercept 10 . Which line has a rate of change that is less than this line's rate of change? (A) \( y=0.5 x+10 \) (B) \( y=-1.5 x+10 \) (C) \( y=-3.5 x+10 \) (D) \( y=-10 x+10 \)

To find the rate of change (slope) of the line passing through the point \( (5, -7.5) \) with a \( y \)-intercept of 10, we can use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using the points \( (5, -7.5) \) and \( (0, 10) \) (the \( y \)-intercept), \[ m = \frac{10 - (-7.5)}{0 - 5} = \frac{10 + 7.5}{-5} = \frac{17.5}{-5} = -3.5. \] Now, we need to compare this slope to the slopes of the given options: (A) \( y = 0.5x + 10 \) has a slope of \( 0.5 \). (B) \( y = -1.5x + 10 \) has a slope of \( -1.5 \). (C) \( y = -3.5x + 10 \) has a slope of \( -3.5 \). (D) \( y = -10x + 10 \) has a slope of \( -10 \). The line with a slope less than \(-3.5\) is \( (D) \, y = -10x + 10\). Thus the answer is: (D) \( y=-10 x+10 \)