To determine the equation of a line with a given slope and a specific point it passes through, we can use the point-slope form of a line equation. The general formula for point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( m \)[/tex] represents the slope of the line, [tex]\((x_1, y_1)\)[/tex] represents a point on the line, and [tex]\((x, y)\)[/tex] are the coordinates of any point on the line. Let's identify the values we need:
- The slope [tex]\( m \)[/tex] is given as 4.
- The point [tex]\((x_1, y_1)\)[/tex] on the line is given as [tex]\((-2, 5)\)[/tex].
Substituting these values into the point-slope formula, we get:
[tex]\[ y - 5 = 4(x - (-2)) \][/tex]
Simplifying the equation within the parentheses:
[tex]\[ y - 5 = 4(x + 2) \][/tex]
Now, let's compare this simplified equation with the given choices:
A. [tex]\( y + 5 = 4(x - 2) \)[/tex]
B. [tex]\( y - 5 = 4(x + 2) \)[/tex]
C. [tex]\( y + 2 = 4(x - 5) \)[/tex]
D. [tex]\( y - 5 = 4(x - 2) \)[/tex]
From our derived equation [tex]\( y - 5 = 4(x + 2) \)[/tex], we see that it matches option B:
[tex]\[ y - 5 = 4(x + 2) \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
