To determine the equation of the line that is parallel to the given line passing through the points [tex]\((0, -3)\)[/tex] and [tex]\((2, 3)\)[/tex] and that also passes through the point [tex]\((-1, -1)\)[/tex], we can proceed step-by-step as follows:
Step 1: Calculate the slope of the given line.
Given points:
- [tex]\((x_1, y_1) = (0, -3)\)[/tex]
- [tex]\((x_2, y_2) = (2, 3)\)[/tex]
The formula to calculate the slope ([tex]\(m\)[/tex]) between two points is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points:
[tex]\[
m = \frac{3 - (-3)}{2 - 0} = \frac{3 + 3}{2} = \frac{6}{2} = 3
\][/tex]
So, the slope of the given line is [tex]\(m = 3\)[/tex].
Step 2: Determine the equation of the line parallel to the given line.
A line parallel to another line will have the same slope. Therefore, the slope of our new line must also be [tex]\(3\)[/tex].
Step 3: Use the point-slope form of the equation of a line.
The point-slope form of a line's equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, our slope [tex]\(m = 3\)[/tex] and our point is [tex]\((-1, -1)\)[/tex].
Substituting the point [tex]\((-1, -1)\)[/tex] into the point-slope form:
[tex]\[
y - (-1) = 3(x - (-1))
\][/tex]
Simplifying, we get:
[tex]\[
y + 1 = 3(x + 1)
\][/tex]
Conclusion:
The equation of the line parallel to the given line passing through the point [tex]\((-1, -1)\)[/tex] is:
[tex]\[
y + 1 = 3(x + 1)
\][/tex]
Thus, among the given options, the correct one is:
[tex]\[
y + 1 = 3(x + 1)
\][/tex]
