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Find the equation of the line passing through the point A(4,7) B(1, -2)

To find the equation of the line passing through the points \( A(4,7) \) and \( B(1, -2) \), we follow these steps:

### Step 1: Calculate the slope \( m \)
The slope of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

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

Substituting the coordinates of points \( A(4,7) \) and \( B(1,-2) \):

\[
m = \frac{-2 - 7}{1 - 4} = \frac{-9}{-3} = 3
\]

### Step 2: Use the point-slope form of the equation
The point-slope form of the equation of a line is:

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

Using the point \( A(4,7) \) and the slope \( m = 3 \):

\[
y - 7 = 3(x - 4)
\]

### Step 3: Simplify the equation
Expand and simplify the equation:

\[
y - 7 = 3x - 12
\]
\[
y = 3x - 12 + 7
\]
\[
y = 3x - 5
\]

### Final Equation of the Line:
The equation of the line passing through the points \( A(4,7) \) and \( B(1,-2) \) is:

\[
y = 3x - 5
\]

Let me know if you have any further questions or need more details.

Here are 5 questions to consider based on this:
1. What is the significance of the slope in the equation of a line?
2. How do we verify that the points A and B satisfy the equation \( y = 3x - 5 \)?
3. How can we convert this line equation to the standard form \( Ax + By = C \)?
4. What is the y-intercept of the line, and how can we find it from the equation?
5. How does changing the points A and B affect the slope of the line?

**Tip**: The slope of a line gives the rate at which the y-coordinate changes per unit increase in the x-coordinate.

Learn how to derive the linear equation of a line passing through points A(4,7) and B(1,-2) with step-by-step instructions. This guide explains using the slope formula and the point-slope form to establish the line's equation, y = 3x - 5. Suitable for students in Grades 8-10.

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