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What is a point-slope equation of the line with slope \(-13\) that goes through the point \((5, 7)\)?

A. \(y + 5 = -13(x + 7)\)
B. \(y + 7 = -13(x + 5)\)
C. \(y - 5 = -13(x - 7)\)
D. [tex]\(y - 7 = -13(x - 5)\)[/tex]

Sagot :

GinnyAnswer
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]
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