What is the area (in square units) of the figure enclosed by the x-axis, the y-axis and the line x + 2y = 6 ?

Options

A. 3
B. 6
C. 9
D. 12

Answer

To find the area of the figure enclosed by the x-axis, the y-axis, and the line given by the equation \(x + 2y = 6\), we can follow these steps:

### Step 1: Convert the equation to intercept form
We start with the equation of the line:
\[ x + 2y = 6 \]
To express this in intercept form, we can rearrange it by isolating \(y\):
\[ 2y = 6 - x \]
\[ y = 3 - \frac{x}{2} \]
Now, we can find the x-intercept and y-intercept.

### Step 2: Find the x-intercept
To find the x-intercept, we set \(y = 0\):
\[ x + 2(0) = 6 \implies x = 6 \]
Thus, the x-intercept is at the point \((6, 0)\).

### Step 3: Find the y-intercept
To find the y-intercept, we set \(x = 0\):
\[ 0 + 2y = 6 \implies 2y = 6 \implies y = 3 \]
Thus, the y-intercept is at the point \((0, 3)\).

### Step 4: Sketch the triangle
Now we have the intercepts:
- x-intercept: \((6, 0)\)
- y-intercept: \((0, 3)\)

These points along with the origin \((0, 0)\) form a right triangle in the first quadrant.

### Step 5: Calculate the area of the triangle
The area \(A\) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, the base is the length along the x-axis (from \((0, 0)\) to \((6, 0)\)), which is \(6\) units, and the height is the length along the y-axis (from \((0, 0)\) to \((0, 3)\)), which is \(3\) units. Therefore:
\[ A = \frac{1}{2} \times 6 \times 3 = \frac{18}{2} = 9 \]

### Final Answer
The area of the figure enclosed by the x-axis, y-axis, and the line \(x + 2y = 6\) is \(9\) square units.

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