what is the area of the triangle formed by graph of x/3 + y/4 = 2 with the x and y-axes?

The equation of the line is given as:

\[
\frac{x}{3} + \frac{y}{4} = 2
\]

To find the area of the triangle formed by this line with the x-axis and y-axis, we need to determine where the line intersects these axes.

### Step 1: Find the x-intercept
The x-intercept occurs where \( y = 0 \). Substituting \( y = 0 \) in the equation:

\[
\frac{x}{3} + \frac{0}{4} = 2 \implies \frac{x}{3} = 2 \implies x = 6
\]

So, the x-intercept is \( (6, 0) \).

### Step 2: Find the y-intercept
The y-intercept occurs where \( x = 0 \). Substituting \( x = 0 \) in the equation:

\[
\frac{0}{3} + \frac{y}{4} = 2 \implies \frac{y}{4} = 2 \implies y = 8
\]

So, the y-intercept is \( (0, 8) \).

### Step 3: Calculate the area of the triangle
The triangle is formed by the points \( (0, 0) \), \( (6, 0) \), and \( (0, 8) \). The formula for the area of a right triangle is:

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

Here, the base is 6 (the x-intercept) and the height is 8 (the y-intercept). Therefore:

\[
\text{Area} = \frac{1}{2} \times 6 \times 8 = 24
\]

Thus, the area of the triangle is **24 square units**.

Would you like further details or have any questions?

Here are 5 related questions to expand the discussion:
1. How do you find the intercepts of a general linear equation?
2. Can this approach be used for other types of lines, like vertical or horizontal lines?
3. What is the geometric meaning of the intercepts in real-life applications?
4. How would the triangle's area change if the line equation had different coefficients?
5. Can you explain the relationship between the slope of the line and the shape of the triangle?

**Tip**: For lines in the form \( \frac{x}{a} + \frac{y}{b} = 1 \), the x-intercept is \( a \) and the y-intercept is \( b \).

What is the area of the triangle formed by the graph of x/3 + y/4 = 2 with the x and y-axes?

Discover how to calculate the area of the triangle formed by the line x/3 + y/4 = 2 and the x- and y-axes. This example demonstrates concepts in linear equations and coordinate geometry, using intercepts to determine triangle dimensions. Ideal for students in Grades 8-10.

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