To find the gradients (slopes) of the lines joining each pair of points, we use the gradient formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
where [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are the coordinates of the two points.
Let's calculate the gradients step by step for each pair of points.
1. (9, 7) and (2, 5):
[tex]\[
m = \frac{5 - 7}{2 - 9} = \frac{-2}{-7} = \frac{2}{7} \approx 0.2857
\][/tex]
2. (2, 5) and (4, 8):
[tex]\[
m = \frac{8 - 5}{4 - 2} = \frac{3}{2} = 1.5
\][/tex]
3. (5, 3) and (0, 0):
[tex]\[
m = \frac{0 - 3}{0 - 5} = \frac{-3}{-5} = \frac{3}{5} = 0.6
\][/tex]
4. (6, 1) and (1, 5):
[tex]\[
m = \frac{5 - 1}{1 - 6} = \frac{4}{-5} = -0.8
\][/tex]
5. (0, 4) and (3, 0):
[tex]\[
m = \frac{0 - 4}{3 - 0} = \frac{-4}{3} \approx -1.3333
\][/tex]
6. (-3, 2) and (4, 4):
[tex]\[
m = \frac{4 - 2}{4 - (-3)} = \frac{2}{7} \approx 0.2857
\][/tex]
7. (2, 3) and (6, -5):
[tex]\[
m = \frac{-5 - 3}{6 - 2} = \frac{-8}{4} = -2
\][/tex]
8. (-4, 3) and (8, -6):
[tex]\[
m = \frac{-6 - 3}{8 - (-4)} = \frac{-9}{12} = \frac{-3}{4} = -0.75
\][/tex]
9. (-4, -4) and (-1, 5):
[tex]\[
m = \frac{5 - (-4)}{-1 - (-4)} = \frac{9}{3} = 3
\][/tex]
10. (7, -2) and (-1, 2):
[tex]\[
m = \frac{2 - (-2)}{-1 - 7} = \frac{4}{-8} = -0.5
\][/tex]
Therefore, the gradients of the lines joining these pairs of points are:
[tex]\[
[0.2857, 1.5, 0.6, -0.8, -1.3333, 0.2857, -2, -0.75, 3, -0.5]
\][/tex]
These values correspond to the calculated gradients for each respective pair of points.
