To find the equation of a line that is parallel to the given line and passes through a specific point, we should follow a systematic approach. Here's a step-by-step solution:
### Step 1: Determine the Slope of the Given Line
First, we need to calculate the slope of the given line, which 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 of 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]\( 3 \)[/tex].
### Step 2: Identify the Slope of the Parallel Line
Since the line we want to find is parallel to the given line, it will have the same slope. Therefore, the slope of our parallel line is also [tex]\( 3 \)[/tex].
### Step 3: Apply the Point-Slope Form of the Line
The point-slope form of the line equation is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is a point on the line, and [tex]\( m \)[/tex] is the slope. We need to write the equation of the line that passes through the point [tex]\((-1, -1)\)[/tex] with a slope of [tex]\( 3 \)[/tex].
Plugging in the slope [tex]\(3\)[/tex] and the point [tex]\((-1, -1)\)[/tex]:
[tex]\[ y - (-1) = 3(x - (-1)) \][/tex]
Simplifying this:
[tex]\[ y + 1 = 3(x + 1) \][/tex]
### Conclusion
The equation of the line that is parallel to the given line and passes through the point [tex]\((-1, -1)\)[/tex] in point-slope form is:
[tex]\[ y + 1 = 3(x + 1) \][/tex]
Among the given options, the correct answer is:
[tex]\[ \boxed{y + 1 = 3(x + 1)} \][/tex]
