Calculate the slope m.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the equations for each pair of points:
a)* Points: $(2, 7)$ and $(4, -2)$
Step 1: Calculate the slope $m$.
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 7}{4 - 2} = \frac{-9}{2}$$
Step 2: Use the point-slope form $y - y_1 = m(x - x_1)$ with point $(2, 7)$.
$$y - 7 = -\frac{9}{2}(x - 2)$$
Step 3: Simplify the equation.
$$y - 7 = -\frac{9}{2}x + 9$$
$$y = -\frac{9}{2}x + 9 + 7$$
$$y = -\frac{9}{2}x + 16$$
The equation of the line is $\boxed{y = -\frac{9}{2}x + 16}$.
b)* Points: $(-1, 5)$ and $(4, -2)$
Step 1: Calculate the slope $m$.
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 5}{4 - (-1)} = \frac{-7}{5}$$
Step 2: Use the point-slope form $y - y_1 = m(x - x_1)$ with point $(-1, 5)$.
$$y - 5 = -\frac{7}{5}(x - (-1))$$
$$y - 5 = -\frac{7}{5}(x + 1)$$
Step 3: Simplify the equation.
$$y - 5 = -\frac{7}{5}x - \frac{7}{5}$$
$$y = -\frac{7}{5}x - \frac{7}{5} + 5$$
$$y = -\frac{7}{5}x - \frac{7}{5} + \frac{25}{5}$$
$$y = -\frac{7}{5}x + \frac{18}{5}$$
The equation of the line is $\boxed{y = -\frac{7}{5}x + \frac{18}{5}}$.
c)* Points: $(-2, 3)$ and $(4, 5)$
Step 1: Calculate the slope $m$.
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 3}{4 - (-2)} = \frac{2}{6} = \frac{1}{3}$$
Step 2: Use the point-slope form $y - y_1 = m(x - x_1)$ with point $(-2, 3)$.
$$y - 3 = \frac{1}{3}(x - (-2))$$
$$y - 3 = \frac{1}{3}(x + 2)$$
Step 3: Simplify the equation.
$$y - 3 = \frac{1}{3}x + \frac{2}{3}$$
$$y = \frac{1}{3}x + \frac{2}{3} + 3$$
$$y = \frac{1}{3}x + \frac{2}{3} + \frac{9}{3}$$
$$y = \frac{1}{3}x + \frac{11}{3}$$
The equation of the line is $\boxed{y = \frac{1}{3}x + \frac{11}{3}}$.
d)* Points: $(-1, -8)$ and $(3, -2)$
Step 1: Calculate the slope $m$.
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - (-8)}{3 - (-1)} = \frac{-2 + 8}{3 + 1} = \frac{6}{4} = \frac{3}{2}$$
Step 2: Use the point-slope form $y - y_1 = m(x - x_1)$ with point $(-1, -8)$.
$$y - (-8) = \frac{3}{2}(x - (-1))$$
$$y + 8 = \frac{3}{2}(x + 1)$$
Step 3: Simplify the equation.
$$y + 8 = \frac{3}{2}x + \frac{3}{2}$$
$$y = \frac{3}{2}x + \frac{3}{2} - 8$$
$$y = \frac{3}{2}x + \frac{3}{2} - \frac{16}{2}$$
$$y = \frac{3}{2}x - \frac{13}{2}$$
The equation of the line is $\boxed{y = \frac{3}{2}x - \frac{13}{2}}$.
e)* Points: $(3, 7)$ and $(8, 4)$
Step 1: Calculate the slope $m$.
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 7}{8 - 3} = \frac{-3}{5}$$
Step 2: Use the point-slope form $y - y_1 = m(x - x_1)$ with point $(3, 7)$.
$$y - 7 = -\frac{3}{5}(x - 3)$$
Step 3: Simplify the equation.
$$y - 7 = -\frac{3}{5}x + \frac{9}{5}$$
$$y = -\frac{3}{5}x + \frac{9}{5} + 7$$
$$y = -\frac{3}{5}x + \frac{9}{5} + \frac{35}{5}$$
$$y = -\frac{3}{5}x + \frac{44}{5}$$
The equation of the line is $\boxed{y = -\frac{3}{5}x + \frac{44}{5}}$.
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