Question

Graph the line that passes through the points left parenthesis, 0, comma, minus, 5, right parenthesis(0,−5) and left parenthesis, 5, comma, 1, right parenthesis(5,1) and determine the equation of the line.

Answer 1

To determine the equation of the line that passes through the points \((0, -5)\) and \((5, 1)\), we follow these steps:

1. **Find the slope (m)** of the line using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Here, \((x_1, y_1) = (0, -5)\) and \((x_2, y_2) = (5, 1)\). Plugging in these values:

\[
m = \frac{1 - (-5)}{5 - 0} = \frac{1 + 5}{5} = \frac{6}{5}
\]

2. **Use the point-slope form of the line**. The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
We can use either point; here we will use \((0, -5)\):

\[
y - (-5) = \frac{6}{5}(x - 0)
\]

Simplifying this:
\[
y + 5 = \frac{6}{5}x
\]
\[
y = \frac{6}{5}x - 5
\]

3. **Convert to slope-intercept form** (if not already in that form):
\[
y = \frac{6}{5}x - 5
\]

So the equation of the line in slope-intercept form is:
\[
y = \frac{6}{5}x - 5
\]

### Graphing the Line
To graph the line, we can plot the two points:

- **Point 1**: \((0, -5)\) (y-intercept)
- **Point 2**: \((5, 1)\)

After plotting these points, draw a straight line through them. This line represents the equation \(y = \frac{6}{5}x - 5\).

### Summary
- The equation of the line is \(y = \frac{6}{5}x - 5\).
- The line passes through the points \((0, -5)\) and \((5, 1)\).