To find the equation of a line that is parallel to a given line and passes through a specific point, we need to find the slope of the given line first and then use this slope in the point-slope form of the equation. Let's go through this step-by-step:
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 formula for 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]
Plugging in the coordinates:
[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. Use the Point-Slope Form of the Line Equation:
The point-slope form of the line equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
We need to find the equation of the line that is parallel to the given line and passes through the point [tex]\((-1, -1)\)[/tex]. Since parallel lines have the same slope, we use the slope [tex]\(m = 3\)[/tex].
3. Plug in the Point and Slope:
We use the point [tex]\((-1, -1)\)[/tex] in the point-slope form:
[tex]\[
y - (-1) = 3(x - (-1))
\][/tex]
Simplifying this equation:
[tex]\[
y + 1 = 3(x + 1)
\][/tex]
Hence, the correct 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 option is:
[tex]\[ y + 1 = 3(x + 1) \][/tex]
