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The problem involves a line given by the equation \(3x - 2y + 6 = 0\). We need to find the area of a right triangle formed by drawing a vertical line from any point on the x-axis to intersect this line, and then by extending it to the y-axis. The area of the triangle needs to be expressed as a function of the x-coordinate, \(x\).

First, let's rearrange the line equation for \(y\):

\[
3x - 2y + 6 = 0 \implies -2y = -3x - 6 \implies y = \frac{3x + 6}{2} = \frac{3}{2}x + 3
\]

The x-intercept occurs where \(y = 0\):

\[
0 = \frac{3}{2}x + 3 \implies x = -2
\]

The y-intercept occurs where \(x = 0\):

\[
y = \frac{3}{2}(0) + 3 = 3
\]

Now, for any \(x\) on the x-axis, we draw a vertical line up to the line \(y = \frac{3}{2}x + 3\). The coordinates of the point of intersection are \((x, \frac{3}{2}x + 3)\).

**Forming the triangle:**
- Base (\(b\)) of the triangle: The base is the horizontal distance from the y-axis to the point \((x, 0)\), which is simply \(|x|\).
- Height (\(h\)) of the triangle: The vertical distance from the x-axis to the line at \(x\), which is \(|\frac{3}{2}x + 3|\).

**Area of the triangle:**
The area \(A\) of a right triangle is given by:

\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]

Here,

\[
A = \frac{1}{2} \times |x| \times \left|\frac{3}{2}x + 3\right|
\]

This expression gives us the area of the triangle in terms of the x-coordinate \(x\). Note that the absolute values are necessary to ensure that the area is always positive, regardless of the position of \(x\) (whether to the left or right of the y-axis).

Would you like a detailed explanation of any part, or have further questions on this topic?

Here are some related questions you might consider:
1. How does changing the slope of the line affect the area of the triangle?
2. What happens to the triangle if the line passes through the origin?
3. Can this method be used to find the area of a triangle formed by other types of lines, such as a curve?
4. How would the area formula change if the line was in a different form, like slope-intercept or point-slope?
5. What is the significance of the absolute value in the area formula?

**Tip:** When dealing with problems involving geometric figures and algebraic equations, sketching the situation can provide a clearer understanding and reveal relationships between the elements of the problem.

When graphing the line whose equation is 3x - 2y + 6 = 0, and drawing a vertical line parallel to the y-axis at any point on the x-axis, a right triangle is generated. Express the area of this triangle as a function of the x-coordinate.

Learn how to express the area of a right triangle generated by a line and a vertical line parallel to the y-axis. This problem covers key algebraic concepts like linear equations and geometric properties of triangles, suitable for students in Grades 10-12.

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