Find an equation for the line that passes through the points $$ (3,-3) $$ and $$ (-5,1) $$.

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UpStudy AI · Accepted Answer

To find the equation of the line that passes through the points $$ (3, -3) $$ and $$ (-5, 1) $$, we can 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 $$ (x_1, y_1) = (3, -3) $$ and $$ (x_2, y_2) = (-5, 1) $$.

2. **Use the point-slope form** of the equation of a line:
   $$
   y - y_1 = m(x - x_1)
   $$
   We can use either of the two points for $$ (x_1, y_1) $$.

Let's perform the calculations step by step.

### Step 1: Calculate the slope (m)

Substituting the coordinates into the slope formula:
$$
m = \frac{1 - (-3)}{-5 - 3} = \frac{1 + 3}{-5 - 3} = \frac{4}{-8} = -\frac{1}{2}
$$

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

Using the point $$ (3, -3) $$:
$$
y - (-3) = -\frac{1}{2}(x - 3)
$$
This simplifies to:
$$
y + 3 = -\frac{1}{2}(x - 3)
$$

Now, let's distribute and rearrange to get the equation in slope-intercept form $$ y = mx + b $$:
$$
y + 3 = -\frac{1}{2}x + \frac{3}{2}
$$
Subtracting 3 from both sides:
$$
y = -\frac{1}{2}x + \frac{3}{2} - 3
$$
$$
y = -\frac{1}{2}x + \frac{3}{2} - \frac{6}{2}
$$
$$
y = -\frac{1}{2}x - \frac{3}{2}
$$

Thus, the equation of the line that passes through the points $$ (3, -3) $$ and $$ (-5, 1) $$ is:
$$
y = -\frac{1}{2}x - \frac{3}{2}
$$
The equation of the line is $$ y = -\frac{1}{2}x - \frac{3}{2} $$.

Find an equation for the line that passes through the points \( (3,-3) \) and \( (-5,1) \).

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