which line has the greatest rate of change?line aline bline cline d

Explanation:

Step1: Define rate of change

The rate of change of a line is its slope, calculated as $\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$. We compare absolute values for "greatest" magnitude.

Step2: Calculate slope of line a

Use points $(0,8)$ and $(8,0)$: $\text{slope}_a = \frac{0 - 8}{8 - 0} = \frac{-8}{8} = -1$ Absolute value: $|-1| = 1$

Step3: Calculate slope of line b

Use points $(0,2)$ and $(3,10)$: $\text{slope}_b = \frac{10 - 2}{3 - 0} = \frac{8}{3} \approx 2.67$ Absolute value: $\left|\frac{8}{3}\right| \approx 2.67$

Step4: Calculate slope of line c

Use points $(0,0)$ and $(10,10)$: $\text{slope}_c = \frac{10 - 0}{10 - 0} = 1$ Absolute value: $|1| = 1$

Step5: Calculate slope of line d

Use points $(0,4)$ and $(8,0)$: $\text{slope}_d = \frac{0 - 4}{8 - 0} = \frac{-4}{8} = -0.5$ Absolute value: $|-0.5| = 0.5$

Step6: Compare absolute slopes

$2.67 > 1 = 1 > 0.5$

Answer:

line b