To determine the equation of the line passing through the points [tex]\((-1, -1)\)[/tex] and [tex]\((-3, 5)\)[/tex], we use the slope-intercept form of a line, which is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] denotes the slope and [tex]\(b\)[/tex] is the y-intercept.
### Step 1: Calculate the Slope ([tex]\(m\)[/tex])
The formula to calculate the slope ([tex]\(m\)[/tex]) between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given coordinates, [tex]\((x_1, y_1) = (-1, -1)\)[/tex] and [tex]\((x_2, y_2) = (-3, 5)\)[/tex], we get:
[tex]\[ m = \frac{5 - (-1)}{-3 - (-1)} \][/tex]
[tex]\[ m = \frac{5 + 1}{-3 + 1} \][/tex]
[tex]\[ m = \frac{6}{-2} \][/tex]
[tex]\[ m = -3 \][/tex]
Thus, the slope of the line is [tex]\(-3\)[/tex].
### Step 2: Calculate the Y-intercept ([tex]\(b\)[/tex])
The y-intercept ([tex]\(b\)[/tex]) is found using one of the points on the line and the slope. The general form is:
[tex]\[ y = mx + b \][/tex]
We can rearrange to solve for [tex]\(b\)[/tex]:
[tex]\[ b = y - mx \][/tex]
Using the point [tex]\((-1, -1)\)[/tex]:
[tex]\[ b = -1 - (-3)(-1) \][/tex]
[tex]\[ b = -1 - 3 \][/tex]
[tex]\[ b = -4 \][/tex]
Thus, the y-intercept is [tex]\(-4\)[/tex].
### Step 3: Form the Equation of the Line
With the slope and y-intercept, we can write the equation of the line:
[tex]\[ y = -3x - 4 \][/tex]
### Conclusion
The equation of the line that passes through the points [tex]\((-1, -1)\)[/tex] and [tex]\((-3, 5)\)[/tex] is:
[tex]\[ y = -3x - 4 \][/tex]
So, the correct choice is:
[tex]\[ y = -3 x - 4 \][/tex]
