Let's find the equation of the line that is parallel to the given line and passes through the point [tex]\((-1, -1)\)[/tex].
1. Determine the Slope of the Given Line:
The given line passes through the points [tex]\((0, -3)\)[/tex] and [tex]\((2, 3)\)[/tex]. The slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the coordinates [tex]\((0, -3)\)[/tex] and [tex]\((2, 3)\)[/tex]:
[tex]\[
m = \frac{3 - (-3)}{2 - 0} = \frac{3 + 3}{2} = \frac{6}{2} = 3
\][/tex]
So, the slope of the given line is 3.
2. Find the Slope of the Parallel Line:
Since the lines are parallel, they have the same slope. Thus, the slope of the new line is also 3.
3. Use the Point-Slope Form Equation:
The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\(m\)[/tex] is the slope (which is 3), and [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes (which is [tex]\((-1, -1)\)[/tex]).
4. Substitute the Slope and Point into the Point-Slope Form:
Substituting [tex]\(m = 3\)[/tex], [tex]\(x_1 = -1\)[/tex], and [tex]\(y_1 = -1\)[/tex] into the point-slope form:
[tex]\[
y - (-1) = 3(x - (-1))
\][/tex]
Simplifying this equation:
[tex]\[
y + 1 = 3(x + 1)
\][/tex]
Thus, the equation of the line that is parallel to the given line and passes through the point [tex]\((-1, -1)\)[/tex] is:
[tex]\[
y + 1 = 3(x + 1)
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{y+1=3(x+1)}
\][/tex]
