Find the equation of the line through the points $$ (-3,-4) $$ and $$ (1,-5) $$. Write your answer in the slope-intercept form $$ y=m x+b $$. $$ y=\square $$

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To find the equation of the line through the points $$ (-3,-4) $$ and $$ (1,-5) $$, we will follow these steps:

1. **Calculate the slope $$ m $$** of the line using the formula:
   $$
   m = \frac{y_2 - y_1}{x_2 - x_1}
   $$
   where $$ (x_1, y_1) = (-3, -4) $$ and $$ (x_2, y_2) = (1, -5) $$.

2. **Use the point-slope form** of the equation of a line to find the equation in slope-intercept form $$ y = mx + b $$.

### Step 1: Calculate the slope $$ m $$

Substituting the coordinates into the slope formula:
$$
m = \frac{-5 - (-4)}{1 - (-3)} = \frac{-5 + 4}{1 + 3} = \frac{-1}{4}
$$

### Step 2: Use the point-slope form

Now that we have the slope $$ m = -\frac{1}{4} $$, we can use one of the points to find the y-intercept $$ b $$. We'll use the point $$ (-3, -4) $$.

The point-slope form of the equation is:
$$
y - y_1 = m(x - x_1)
$$
Substituting $$ m $$, $$ x_1 $$, and $$ y_1 $$:
$$
y - (-4) = -\frac{1}{4}(x - (-3))
$$
This simplifies to:
$$
y + 4 = -\frac{1}{4}(x + 3)
$$

Now, distribute the slope:
$$
y + 4 = -\frac{1}{4}x - \frac{3}{4}
$$

Next, isolate $$ y $$:
$$
y = -\frac{1}{4}x - \frac{3}{4} - 4
$$
Convert $$ -4 $$ to a fraction with a denominator of 4:
$$
-4 = -\frac{16}{4}
$$
So,
$$
y = -\frac{1}{4}x - \frac{3}{4} - \frac{16}{4}
$$
Combine the constants:
$$
y = -\frac{1}{4}x - \frac{19}{4}
$$

### Final Answer

The equation of the line in slope-intercept form is:
$$
y = -\frac{1}{4}x - \frac{19}{4}
$$
The equation of the line is $$ y = -\frac{1}{4}x - \frac{19}{4} $$.

Find the equation of the line through the points \( (-3,-4) \) and \( (1,-5) \). Write your answer in the slope-intercept form \( y=m x+b \). \( y=\square \)

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