Which point-slope equation represents a line that passes through \((3, -2)\) with a slope of \(-\frac{4}{5}\)?

A. \(y - 3 = -\frac{4}{5}(x + 2)\)
B. \(y - 2 = \frac{4}{5}(x - 3)\)
C. \(y + 2 = -\frac{4}{5}(x - 3)\)
D. [tex]\(y + 3 = \frac{4}{5}(x + 2)\)[/tex]



Answer :

To determine which point-slope form equation represents the line that passes through the point \((3, -2)\) with a slope of \(-\frac{4}{5}\), let's follow through the point-slope form of a linear equation.

The point-slope form is expressed as:

[tex]\[ y - y_1 = m(x - x_1) \][/tex]

where \( (x_1, y_1) \) is a given point on the line, and \( m \) is the slope of the line.

Given:
- The point \((3, -2)\) means \( x_1 = 3 \) and \( y_1 = -2 \).
- The slope \( m = -\frac{4}{5} \).

Substituting these values into the point-slope form, we get:

[tex]\[ y - (-2) = -\frac{4}{5}(x - 3) \][/tex]

Simplify the left side of the equation:

[tex]\[ y + 2 = -\frac{4}{5}(x - 3) \][/tex]

So, the point-slope equation that correctly represents the line passing through the point \((3, -2)\) with a slope of \(-\frac{4}{5}\) is:

[tex]\[ y + 2 = -\frac{4}{5}(x - 3) \][/tex]

Therefore, the correct answer is the third option:

[tex]\[ \boxed{y + 2 = -\frac{4}{5}(x - 3)} \][/tex]