To determine the equation of a line passing through two points, [tex]\((-3, -2)\)[/tex] and [tex]\( (2, -7) \)[/tex], we need to follow these steps:
1. Calculate the slope (m) of the line: The slope [tex]\( m \)[/tex] is given by the 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.
For the given points [tex]\((-3, -2)\)[/tex] and [tex]\( (2, -7) \)[/tex]:
[tex]\[
x_1 = -3, \quad y_1 = -2, \quad x_2 = 2, \quad y_2 = -7
\][/tex]
Plugging these values into the slope formula:
[tex]\[
m = \frac{-7 - (-2)}{2 - (-3)} = \frac{-7 + 2}{2 + 3} = \frac{-5}{5} = -1.0
\][/tex]
2. Find the y-intercept (b) of the line: The equation of the line in slope-intercept form is [tex]\( y = mx + b \)[/tex].
To find the y-intercept [tex]\( b \)[/tex], we can use the slope [tex]\( m \)[/tex] and one of the points [tex]\((x_1, y_1)\)[/tex].
Use the point [tex]\((-3, -2)\)[/tex] and the slope [tex]\( m = -1.0 \)[/tex]:
[tex]\[
y = mx + b \quad \Rightarrow \quad -2 = -1.0(-3) + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
-2 = 3.0 + b \quad \Rightarrow \quad b = -2 - 3.0 = -5.0
\][/tex]
3. Write the equation of the line: Now that we have the slope [tex]\( m = -1.0 \)[/tex] and the y-intercept [tex]\( b = -5.0 \)[/tex], we can write the equation of the line:
[tex]\[
y = -1.0x + (-5.0)
\][/tex]
Simplifying, the equation is:
[tex]\[
y = -1.0x - 5.0
\][/tex]
Therefore, the correct equation of the line passing through the points [tex]\((-3, -2)\)[/tex] and [tex]\( (2, -7) \)[/tex] is:
[tex]\[
\boxed{y = -1.0x - 5.0}
\][/tex]
In this solution, it appears there's a mistake in the provided attempt with [tex]\( y = -x + 5 \)[/tex]; the correct answer should be [tex]\( y = -1.0x - 5.0 \)[/tex].
