To determine the equation of a line in the point-slope form, we will use the given slope and the coordinates of the given point. The point-slope form of a line is defined as:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( m \)[/tex] is the slope of the line, and [tex]\((x_1, y_1)\)[/tex] are the coordinates of a point on the line.
Given:
- Slope [tex]\(\frac{2}{5}\)[/tex]
- Point [tex]\((-4, -7)\)[/tex]
We substitute these values into the point-slope form equation.
1. Identify [tex]\( m \)[/tex], [tex]\( x_1 \)[/tex], and [tex]\( y_1 \)[/tex]:
[tex]\[ m = \frac{2}{5}, \quad x_1 = -4, \quad y_1 = -7 \][/tex]
2. Substitute these values into the point-slope form equation:
[tex]\[ y - (-7) = \frac{2}{5} (x - (-4)) \][/tex]
3. Simplify the equation:
Since [tex]\( y - (-7) \)[/tex] becomes [tex]\( y + 7 \)[/tex] and [tex]\( x - (-4) \)[/tex] becomes [tex]\( x + 4 \)[/tex], we get:
[tex]\[ y + 7 = \frac{2}{5} (x + 4) \][/tex]
Thus, the point-slope form of the line with the given slope [tex]\(\frac{2}{5}\)[/tex] and passing through the point [tex]\((-4, -7)\)[/tex] is:
[tex]\[ y + 7 = \frac{2}{5}(x + 4) \][/tex]
Now we compare it with the given options:
1. [tex]\( y + 7 = \frac{2}{5}(x + 4) \)[/tex]
2. [tex]\( y - 4 = \frac{2}{5}(x - 7) \)[/tex]
3. [tex]\( y + 4 = \frac{2}{5}(x + 7) \)[/tex]
4. [tex]\( y - 7 = \frac{2}{5}(x - 4) \)[/tex]
The correct option is:
[tex]\[ y + 7 = \frac{2}{5}(x + 4) \][/tex]
Therefore, the correct answer is the first option.
