Sure, let's determine the equation of the line that passes through the points [tex]\((-3, 1)\)[/tex] and [tex]\((-2, -5)\)[/tex].
### Step 1: Find the slope (m) of the line.
The slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substitute the points [tex]\((-3, 1)\)[/tex] and [tex]\((-2, -5)\)[/tex] into the formula:
[tex]\[
m = \frac{-5 - 1}{-2 - (-3)} = \frac{-5 - 1}{-2 + 3} = \frac{-6}{1} = -6
\][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(-6\)[/tex].
### Step 2: Find the y-intercept (b) of the line.
The y-intercept [tex]\(b\)[/tex] can be found using the point-slope form of the line equation [tex]\(y = mx + b\)[/tex]. We can use one of the points and the calculated slope to find [tex]\(b\)[/tex].
Let's use the point [tex]\((-3, 1)\)[/tex]. Substitute [tex]\(x = -3\)[/tex], [tex]\(y = 1\)[/tex], and [tex]\(m = -6\)[/tex] into the equation [tex]\(y = mx + b\)[/tex]:
[tex]\[
1 = -6(-3) + b
\][/tex]
Simplify:
[tex]\[
1 = 18 + b
\][/tex]
Subtract 18 from both sides to solve for [tex]\(b\)[/tex]:
[tex]\[
b = 1 - 18
\][/tex]
[tex]\[
b = -17
\][/tex]
So, the y-intercept [tex]\(b\)[/tex] is [tex]\(-17\)[/tex].
### Step 3: Write the equation of the line.
Now that we have the slope [tex]\(m = -6\)[/tex] and the y-intercept [tex]\(b = -17\)[/tex], we can write the equation of the line:
[tex]\[
y = -6x - 17
\][/tex]
### Step 4: Compare with the given options.
Let's match this equation with the provided options:
a. [tex]\(y=6x-17\)[/tex]
b. [tex]\(y=-6x-17\)[/tex]
c. [tex]\(y=-6x+17\)[/tex]
d. [tex]\(y=-6x+\frac{1}{17}\)[/tex]
The correct equation is:
[tex]\[
y = -6x - 17
\][/tex]
Therefore, the answer is:
b. [tex]\(y = -6x - 17\)[/tex]
