Choose the equation below that represents the line passing through the point [tex](-2, -3)[/tex] with a slope of -6.

A. [tex]y = -6x - 15[/tex]
B. [tex]y = -6x - 20[/tex]
C. [tex]y = -6x + 15[/tex]
D. [tex]y = -6x + 20[/tex]



Answer :

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.