through $$ (1,-3) $$ and $$ (4,-2) $$ A. Write the point-slope form of the equation of the line through the given points: B. $$ y=-\frac{1}{3}(x+1) $$ B. $$ y+1=\frac{1}{3}(x+3) $$ C. $$ y-1=-\frac{1}{3}(x+3) $$ D. $$ y+3=\frac{1}{3}(x-1) $$

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To determine the correct point-slope form of the equation of the line passing through the points $$(1, -3)$$ and $$(4, -2)$$, follow these steps:

### Step 1: Calculate the Slope ($$m$$)

The slope $$m$$ of a line passing through two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is given by:

$$
m = \frac{y_2 - y_1}{x_2 - x_1}
$$

Plugging in the given points:

$$
m = \frac{-2 - (-3)}{4 - 1} = \frac{1}{3}
$$

### Step 2: Use the Point-Slope Formula

The point-slope form of a line is:

$$
y - y_1 = m(x - x_1)
$$

Using the point $$(1, -3)$$:

$$
y - (-3) = \frac{1}{3}(x - 1) \
y + 3 = \frac{1}{3}(x - 1)
$$

This matches **Option D**.

### Verification of Other Options:

- **Option B**: $$ y = -\frac{1}{3}(x + 1) $$  
  *Incorrect slope and form.*

- **Option B**: $$ y + 1 = \frac{1}{3}(x + 3) $$  
  *Incorrect point used.*

- **Option C**: $$ y - 1 = -\frac{1}{3}(x + 3) $$  
  *Incorrect slope and point.*

- **Option D**: $$ y + 3 = \frac{1}{3}(x - 1) $$  
  *Correct slope and uses the point $$(1, -3)$$.*

### **Answer: D. $$ y + 3 = \frac{1}{3}(x - 1) $$**
The correct point-slope form of the equation is $$ y + 3 = \frac{1}{3}(x - 1) $$, which is option D.

through and
A. Write the point-slope form of the equation of the line through the given points:
B.
B.
C.
D.

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