To determine which point-slope equation represents a line that passes through the point [tex]\( (3, -2) \)[/tex] with a slope of [tex]\( -\frac{4}{5} \)[/tex], let's analyze each given equation step-by-step.
The general form of the point-slope equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes, and [tex]\(m\)[/tex] is the slope. For our specific problem, the point is [tex]\((3, -2)\)[/tex] and the slope [tex]\(m\)[/tex] is [tex]\( -\frac{4}{5} \)[/tex].
### Analyze Each Equation
Equation 1: [tex]\( y - 3 = -\frac{4}{5}(x + 2) \)[/tex]
Here, the point provided by this equation seems to be taken as [tex]\((x_1, y_1) = (-2, 3)\)[/tex]
but our point is [tex]\((3, -2)\)[/tex]. This does not match with the given point and slope.
Equation 2: [tex]\( y - 2 = \frac{4}{5}(x - 3) \)[/tex]
This implies:
[tex]\[ y - 2 = \frac{4}{5}(x - 3) \][/tex]
Comparing this with the general form [tex]\( y - y_1 = m(x - x_1) \)[/tex]:
- [tex]\( y_1 = 2 \)[/tex]
- [tex]\( m = \frac{4}{5} \)[/tex]
- [tex]\( x_1 = 3 \)[/tex]
This does not match the point [tex]\((3, -2)\)[/tex] and the slope [tex]\( -\frac{4}{5} \)[/tex].
Equation 3: [tex]\( y + 2 = -\frac{4}{5}(x - 3) \)[/tex]
This implies:
[tex]\[ y + 2 = -\frac{4}{5}(x - 3) \][/tex]
Comparing this with the general form [tex]\( y - y_1 = m(x - x_1) \)[/tex]:
- [tex]\( y_1 = -2 \)[/tex]
- [tex]\( m = -\frac{4}{5} \)[/tex]
- [tex]\( x_1 = 3 \)[/tex]
This matches with the given point [tex]\((3, -2)\)[/tex] and the slope [tex]\( -\frac{4}{5} \)[/tex].
Equation 4: [tex]\( y + 3 = \frac{4}{5}(x + 2) \)[/tex]
The point provided by this equation seems to be taken as [tex]\((x_1, y_1) = (-2, -3)\)[/tex] but our point is [tex]\( (3, -2) \)[/tex]. This does not match with the given point and slope.
### Correct Equation
Based on our analysis, the equation that correctly represents the line passing through [tex]\( (3, -2) \)[/tex] with a slope of [tex]\( -\frac{4}{5} \)[/tex] is:
[tex]\[ y + 2 = -\frac{4}{5}(x - 3) \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{3} \][/tex]
