Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

Find the gradients of the lines joining the following pairs of points.

[tex]\[
\begin{tabular}{lr}
1. (9, 7), (2, 5) & 2. (2, 5), (4, 8) \\
3. (5, 3), (0, 0) & 4. (6, 1), (1, 5) \\
5. (0, 4), (3, 0) & 6. (-3, 2), (4, 4) \\
7. (2, 3), (6, -5) & 8. (-4, 3), (8, -6) \\
9. (-4, -4), (-1, 5) & 10. (7, -2), (-1, 2) \\
\end{tabular}
\][/tex]


Sagot :

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.