To find the point-slope form of the equation of a line given a slope and a point it passes through, we use the point-slope form formula:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Where:
- \( (x_1, y_1) \) is a point on the line,
- \( m \) is the slope of the line.
Given:
- Slope \( m = -13 \)
- Point \( (x_1, y_1) = (5, 7) \)
Substituting the given values into the formula:
[tex]\[ y - 7 = -13(x - 5) \][/tex]
Let's check this against the given options:
A. \( y+5=-13(x+7) \)
This does not match because it has \( y+5 \) and \( x+7 \) instead of \( y-7 \) and \( x-5 \).
B. \( y+7=-13(x+5) \)
This also does not match because it has \( y+7 \) and \( x+5 \) instead of \( y-7 \) and \( x-5 \).
C. \( y-5=-13(x-7) \)
This does not match either because it has \( y-5 \) and \( x-7 \) instead of \( y-7 \) and \( x-5 \).
D. \( y-7=-13(x-5) \)
This matches the equation we derived.
Thus, the correct answer is:
[tex]\[ y - 7 = -13(x - 5) \][/tex]
Hence, the correct option is:
D. [tex]\( y-7=-13(x-5) \)[/tex]
