To find the equation of the line that passes through the points [tex]\((-3, -7.5)\)[/tex] and [tex]\( (2, -5) \)[/tex], follow these steps:
1. Identify the coordinates of the points:
- Point 1 [tex]\((x_1, y_1) = (-3, -7.5)\)[/tex]
- Point 2 [tex]\((x_2, y_2) = (2, -5)\)[/tex]
2. Calculate the slope ([tex]\(m\)[/tex]) of the line using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points into the formula:
[tex]\[
m = \frac{-5 - (-7.5)}{2 - (-3)} = \frac{-5 + 7.5}{2 + 3} = \frac{2.5}{5} = 0.5
\][/tex]
3. Use the point-slope form of the linear equation to find the y-intercept ([tex]\(b\)[/tex]).
The point-slope form is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Rearranging to solve for [tex]\(y\)[/tex]:
[tex]\[
y = mx + b
\][/tex]
To find [tex]\(b\)[/tex], use one of the points, say [tex]\((-3, -7.5)\)[/tex], and substitute [tex]\(m = 0.5\)[/tex]:
[tex]\[
-7.5 = 0.5 \cdot (-3) + b
\][/tex]
Simplify:
[tex]\[
-7.5 = -1.5 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = -7.5 + 1.5 = -6
\][/tex]
4. Write the final equation of the line:
Substituting [tex]\(m = 0.5\)[/tex] and [tex]\(b = -6\)[/tex] into [tex]\(y = mx + b\)[/tex]:
[tex]\[
y = 0.5x - 6
\][/tex]
Therefore, the equation of the line passing through the points [tex]\((-3, -7.5)\)[/tex] and [tex]\((2, -5)\)[/tex] is:
[tex]\[
y = 0.5x - 6
\][/tex]
