To find the equation of the line that is parallel to a given line and passes through a specific point, we need to follow a systematic approach:
### 1. Determine the Slope of the Given Line:
First, calculate the slope of the given line that passes through the points [tex]\((0, -3)\)[/tex] and [tex]\((2, 3)\)[/tex].
The slope formula is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the coordinates [tex]\((x_1, y_1) = (0, -3)\)[/tex] and [tex]\((x_2, y_2) = (2, 3)\)[/tex]:
[tex]\[ m = \frac{3 - (-3)}{2 - 0} \][/tex]
[tex]\[ m = \frac{3 + 3}{2} \][/tex]
[tex]\[ m = \frac{6}{2} \][/tex]
[tex]\[ m = 3 \][/tex]
### 2. Identify the Slope of the Parallel Line:
Since parallel lines have the same slope, the slope of the new line is also [tex]\(3\)[/tex].
### 3. Use the Point-Slope Form to Find the Equation:
The point-slope form of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
We need the equation of the line passing through the point [tex]\((-1, -1)\)[/tex] with the slope [tex]\(3\)[/tex].
Plug in [tex]\(m = 3\)[/tex], [tex]\((x_1, y_1) = (-1, -1)\)[/tex]:
[tex]\[ y - (-1) = 3(x - (-1)) \][/tex]
[tex]\[ y + 1 = 3(x + 1) \][/tex]
### Conclusion:
Thus, the equation of the line in point-slope form that is parallel to the given line and passes through the point [tex]\((-1, -1)\)[/tex] is:
[tex]\[ y + 1 = 3(x + 1) \][/tex]
