Determine the equation of the line 𝐴𝐶.

To determine the equation of the line passing through points \( A(2, 1) \) and \( C(1, -4) \), we use the point-slope form of the equation of a line:

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

### Step 1: Calculate the Slope
First, we need to find the slope \( m \) of the line \( AC \) using the formula:

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

Here, the points are:
- \( A(2, 1) \) where \( x_1 = 2 \) and \( y_1 = 1 \)
- \( C(1, -4) \) where \( x_2 = 1 \) and \( y_2 = -4 \)

Calculate the slope:

\[
m = \frac{-4 - 1}{1 - 2} = \frac{-5}{-1} = 5
\]

So, the slope \( m = 5 \).

### Step 2: Use the Point-Slope Form
Now, substitute the slope \( m = 5 \) and the coordinates of point \( A(2, 1) \) into the point-slope form equation:

\[
y - 1 = 5(x - 2)
\]

### Step 3: Simplify to the Slope-Intercept Form
To convert this to the slope-intercept form \( y = mx + b \):

\[
y - 1 = 5x - 10
\]
\[
y = 5x - 9
\]

So, the equation of the line \( AC \) is:

\[
y = 5x - 9
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you convert a point-slope form equation to the slope-intercept form?
2. What is the importance of the y-intercept in the equation of a line?
3. How do you verify if a point lies on a given line?
4. How do you determine the equation of a line parallel to \( AC \) and passing through a different point?
5. Can you find the x-intercept of the line \( AC \) from its equation?

**Tip:** Always double-check the coordinates and calculations when determining the slope and equation of a line to avoid simple errors.

Learn how to determine the equation of line AC passing through points (2, 1) and (1, -4) using the point-slope form of a line equation. This problem covers key concepts in geometry and coordinate geometry, including the slope formula and point-slope form. Suitable for students in Grades 7-9.

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