To determine the equation of a line given a slope and a point the line passes through, we can use the point-slope form of the equation of a line. The point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line.
Given:
- Slope [tex]\( m = 5 \)[/tex]
- Point [tex]\((-2, 4)\)[/tex]
Plugging these values into the point-slope form:
[tex]\[ y - 4 = 5(x - (-2)) \][/tex]
Simplify the equation:
[tex]\[ y - 4 = 5(x + 2) \][/tex]
Distribute the slope on the right-hand side:
[tex]\[ y - 4 = 5x + 10 \][/tex]
Now, isolate [tex]\( y \)[/tex] to convert the equation into the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y = 5x + 10 + 4 \][/tex]
Combine the constant terms:
[tex]\[ y = 5x + 14 \][/tex]
Therefore, the correct equation of the line is:
[tex]\[ \boxed{y = 5x + 14} \][/tex]
This corresponds to choice [tex]\( d \)[/tex].
