To find the equation of the line passing through the points [tex]\((7, -8)\)[/tex] and [tex]\((-1, -5)\)[/tex], we will follow a step-by-step approach.
1. Identify the Coordinates:
We are given two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
(x_1, y_1) = (7, -8)
\][/tex]
[tex]\[
(x_2, y_2) = (-1, -5)
\][/tex]
2. Calculate the Slope:
The slope (m) of a line passing through two points is found using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the values from our points:
[tex]\[
m = \frac{-5 - (-8)}{-1 - 7} = \frac{-5 + 8}{-1 - 7} = \frac{3}{-8} = -0.375
\][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(-0.375\)[/tex].
3. Find the Y-Intercept (b):
To find the y-intercept, we use the slope-intercept form of the equation of a line:
[tex]\[
y = mx + b
\][/tex]
Substituting one of the points and the slope into this formula to solve for [tex]\(b\)[/tex]:
Using the point [tex]\((x_1, y_1) = (7, -8)\)[/tex]:
[tex]\[
-8 = -0.375 \times 7 + b
\][/tex]
Calculate the product:
[tex]\[
-8 = -2.625 + b
\][/tex]
Solve for [tex]\(b\)[/tex]:
[tex]\[
b = -8 + 2.625 = -5.375
\][/tex]
So, the y-intercept [tex]\(b\)[/tex] is [tex]\(-5.375\)[/tex].
4. Write the Equation:
Now that we have the slope and y-intercept, we can write the equation of the line in slope-intercept form:
[tex]\[
y = -0.375x - 5.375
\][/tex]
Therefore, the equation of the line in slope-intercept form is:
[tex]\[
y = -0.375x - 5.375
\][/tex]
