To determine which equation represents a line that passes through the point [tex]\((-2, 4)\)[/tex] and has a slope of [tex]\(\frac{2}{5}\)[/tex], we should use the point-slope form of the equation of a line. The point-slope form is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where:
- [tex]\( (x_1, y_1) \)[/tex] is a point on the line,
- [tex]\( m \)[/tex] is the slope of the line.
Given:
- [tex]\( (x_1, y_1) = (-2, 4) \)[/tex]
- [tex]\( m = \frac{2}{5} \)[/tex]
Substituting these values into the point-slope form, we get:
[tex]\[ y - 4 = \frac{2}{5}[x - (-2)] \][/tex]
[tex]\[ y - 4 = \frac{2}{5}(x + 2) \][/tex]
Now, let's compare this with the provided options:
1. [tex]\( y-4=\frac{2}{5}(x+2) \)[/tex]
2. [tex]\( y+4=\frac{2}{5}(x-2) \)[/tex]
3. [tex]\( y+2=\frac{2}{5}(x-4) \)[/tex]
4. [tex]\( y-2=\frac{2}{5}(x+4) \)[/tex]
The equation [tex]\( y - 4 = \frac{2}{5}(x + 2) \)[/tex] matches the first option:
[tex]\[ y-4=\frac{2}{5}(x+2) \][/tex]
Therefore, the correct equation that represents a line passing through [tex]\((-2, 4)\)[/tex] with a slope of [tex]\(\frac{2}{5}\)[/tex] is:
[tex]\[ \boxed{y-4=\frac{2}{5}(x+2)} \][/tex]
