To find the equation of the line parallel to the one that passes through the given points [tex]\((0, -3)\)[/tex] and [tex]\((2, 3)\)[/tex] and 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
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] can be calculated using the slope formula:
[tex]\[m = \frac{y_2 - y_1}{x_2 - x_1}\][/tex]
Substituting the coordinates of the given points [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]
Step 2: Use the Point-Slope Form of the Equation of a Line
The point-slope form of a line with slope [tex]\(m\)[/tex] passing through a point [tex]\((x_1, y_1)\)[/tex] is:
[tex]\[y - y_1 = m(x - x_1)\][/tex]
In this case, the slope is [tex]\(3\)[/tex] (since the lines are parallel, they have the same slope), and the line passes through the point [tex]\((-1, -1)\)[/tex].
So, we have:
[tex]\[y - (-1) = 3(x - (-1))\][/tex]
Step 3: Simplify the Equation
Simplify the left and right sides of the equation:
[tex]\[y + 1 = 3(x + 1)\][/tex]
Conclusion
The equation of the line in point-slope form, parallel to the given line and passing through the point [tex]\((-1, -1)\)[/tex], is:
[tex]\[y + 1 = 3(x + 1)\][/tex]
Thus, the correct answer is:
[tex]\[y + 1 = 3(x + 1)\][/tex]
