Question

Graph the line that passes through the points left parenthesis, 4, comma, minus, 1, right parenthesis(4,−1) and left parenthesis, 8, comma, 2, right parenthesis(8,2) and determine the equation of the line.
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Answer 1

To graph the line that passes through the points \( (4, -1) \) and \( (8, 2) \) and to determine its equation, we can follow these steps:

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

The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \( (4, -1) \) and \( (8, 2) \):
- \( x_1 = 4 \), \( y_1 = -1 \)
- \( x_2 = 8 \), \( y_2 = 2 \)

Substituting these values into the formula:

\[
m = \frac{2 - (-1)}{8 - 4} = \frac{2 + 1}{8 - 4} = \frac{3}{4}
\]

### Step 2: Use the point-slope form to find the equation of the line

The point-slope form of the equation of a line is:

\[
y - y_1 = m(x - x_1)
\]

We can use either point; let's use \( (4, -1) \):

\[
y - (-1) = \frac{3}{4}(x - 4)
\]

Simplifying gives:

\[
y + 1 = \frac{3}{4}x - 3
\]

Subtracting 1 from both sides:

\[
y = \frac{3}{4}x - 4
\]

### Step 3: Convert to slope-intercept form (if necessary)

The equation \( y = \frac{3}{4}x - 4 \) is already in slope-intercept form, where:
- The slope \( m = \frac{3}{4} \)
- The y-intercept \( b = -4 \)

### Step 4: Graph the line

1. **Plot the points**: \( (4, -1) \) and \( (8, 2) \).
2. **Draw the line** that connects these two points, ensuring it extends in both directions.

### Final Equation

The equation of the line that passes through the points \( (4, -1) \) and \( (8, 2) \) is:

\[
y = \frac{3}{4}x - 4
\]

You can plot this equation by choosing some values for \( x \) and calculating the corresponding \( y \) values, or by starting from the y-intercept \(-4\) and using the slope to find additional points.