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]
