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The given line passes through the points (0,-3) and (2,3).

What is the equation, in point-slope form, of the line that is parallel to the given line and passes through the point (-1,-1)?

A. [tex]\( y + 1 = 3(x + 1) \)[/tex]
B. [tex]\( y + 1 = -3(x + 1) \)[/tex]
C. [tex]\( y + 1 = \frac{1}{3}(x + 1) \)[/tex]


Sagot :

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]