To determine the equation of the line passing through the point [tex]\((-2, -3)\)[/tex] with a slope of [tex]\(-6\)[/tex], we will use the point-slope form of the equation of a line. The general form is:
[tex]\[ y = mx + b \][/tex]
where [tex]\(m\)[/tex] is the slope, and [tex]\(b\)[/tex] is the y-intercept.
### Step-by-Step Solution:
1. Identify the given values:
- Slope [tex]\(m = -6\)[/tex]
- Point [tex]\((-2, -3)\)[/tex]
2. Substitute the given point [tex]\((x, y)\)[/tex] and the slope [tex]\(m\)[/tex] into the equation [tex]\( y = mx + b \)[/tex] to find [tex]\(b\)[/tex]:
[tex]\[
-3 = -6 \cdot (-2) + b
\][/tex]
3. Calculate the product of the slope and [tex]\(x\)[/tex]:
[tex]\[
-6 \cdot (-2) = 12
\][/tex]
4. Substitute back into the equation to solve for [tex]\(b\)[/tex]:
[tex]\[
-3 = 12 + b
\][/tex]
5. Isolate [tex]\(b\)[/tex] by subtracting 12 from both sides:
[tex]\[
b = -3 - 12
\][/tex]
[tex]\[
b = -15
\][/tex]
6. Write the equation of the line using the found y-intercept [tex]\(b = -15\)[/tex] and slope [tex]\(m = -6\)[/tex]:
[tex]\[
y = -6x - 15
\][/tex]
From the options provided, the correct equation representing the line passing through the point [tex]\((-2, -3)\)[/tex] with a slope of [tex]\(-6\)[/tex] is:
[tex]\[ y = -6x - 15 \][/tex]
Therefore, the correct answer is:
- [tex]\( y = -6x - 15 \)[/tex]
This corresponds to option 1.
