To determine which equation represents the line passing through the point [tex]\((-2, -3)\)[/tex] with a slope of [tex]\(-6\)[/tex], we can use the point-slope form of a linear equation. The point-slope form is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line, and [tex]\(m\)[/tex] is the slope of the line. Let's apply the given point [tex]\((-2, -3)\)[/tex] and the slope [tex]\(-6\)[/tex].
1. Substitute the given point and slope into the point-slope form:
[tex]\[ y - (-3) = -6(x - (-2)) \][/tex]
2. Simplify the equation:
[tex]\[ y + 3 = -6(x + 2) \][/tex]
Hence, the equation that represents the line passing through the point [tex]\((-2, -3)\)[/tex] with a slope of [tex]\(-6\)[/tex] is:
[tex]\[ y + 3 = -6(x + 2) \][/tex]
We can now match this equation with the given options:
1. [tex]\(y+3=-6(x+2)\)[/tex]
2. [tex]\(y+2=6(x+3)\)[/tex]
3. [tex]\(y-3=-6(x-2)\)[/tex]
4. [tex]\(y-2=6(x-3)\)[/tex]
The correct equation is option 1:
[tex]\[ y+3=-6(x+2) \][/tex]
Therefore, the answer is the first option.
