To find the equation of a line given a point \((3, -2)\) and a slope of \(-\frac{4}{3}\), we can use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.
Plugging in the given point and slope:
- \(x_1 = 3\)
- \(y_1 = -2\)
- \(m = -\frac{4}{3}\)
Substituting these values into the point-slope form:
\[ y - (-2) = -\frac{4}{3}(x - 3) \]
This simplifies to:
\[ y + 2 = -\frac{4}{3}(x - 3) \]
Next, distribute \(-\frac{4}{3}\):
\[ y + 2 = -\frac{4}{3}x + 4 \]
Now, isolate \(y\) by subtracting 2 from both sides:
\[ y = -\frac{4}{3}x + 4 - 2 \]
This simplifies to:
\[ y = -\frac{4}{3}x + 2 \]
Thus, the equation of the line is:
\[ y = -\frac{4}{3}x + 2 \]