To derive the equation of the line passing through the points [tex]\((-3, -7.5)\)[/tex] and [tex]\( (2, -5) \)[/tex], let's follow these steps:
1. Find the slope (m) of the line:
The formula to calculate the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For the points [tex]\((-3, -7.5)\)[/tex] and [tex]\( (2, -5)\)[/tex],
[tex]\[
m = \frac{-5 - (-7.5)}{2 - (-3)} = \frac{-5 + 7.5}{2 + 3} = \frac{2.5}{5} = 0.5
\][/tex]
2. Find the y-intercept (b):
Using the point-slope form of the linear equation, [tex]\( y - y_1 = m(x - x_1) \)[/tex], we can rearrange this to the slope-intercept form [tex]\( y = mx + b \)[/tex].
Substitute one of the points and the slope [tex]\(m\)[/tex] into this formula:
[tex]\[
y - y_1 = m(x - x_1) \rightarrow y + 7.5 = 0.5(x + 3)
\][/tex]
To find the y-intercept ([tex]\(b\)[/tex]), we rearrange to:
[tex]\[
y = 0.5x + b
\][/tex]
Substitute the point [tex]\((-3, -7.5)\)[/tex] into [tex]\(y = 0.5x + b\)[/tex]:
[tex]\[
-7.5 = 0.5(-3) + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
-7.5 = -1.5 + b \rightarrow b = -7.5 + 1.5 = -6
\][/tex]
3. Write the equation of the line:
Substituting the slope [tex]\(m = 0.5\)[/tex] and the y-intercept [tex]\(b = -6\)[/tex] in the slope-intercept form [tex]\( y = mx + b \)[/tex], the equation of the line becomes:
[tex]\[
y = 0.5x - 6
\][/tex]
Thus, 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]
