Area of triangle formed by corrdinate axes and straight line `y = 3x - 14 ` is

Options

A. `(196)/(3)`
B. `( 49)/(3)`
C. `(98)/(3)`
D. `(3)/(98)`

Answer

To find the area of the triangle formed by the coordinate axes and the straight line given by the equation \( y = 3x - 14 \), we can follow these steps:

### Step 1: Find the x-intercept
To find the x-intercept, we set \( y = 0 \) in the equation of the line.

\[ 0 = 3x - 14 \]

Solving for \( x \):

\[ 3x = 14 \implies x = \frac{14}{3} \]

### Step 2: Find the y-intercept
Next, we find the y-intercept by setting \( x = 0 \) in the equation of the line.

\[ y = 3(0) - 14 = -14 \]

### Step 3: Identify the vertices of the triangle
The triangle is formed by the points:
- The origin \( O(0, 0) \)
- The x-intercept \( B\left(\frac{14}{3}, 0\right) \)
- The y-intercept \( A(0, -14) \)

### Step 4: Calculate the area of the triangle
The area \( A \) of a triangle formed by the coordinate axes and a straight line can be calculated using the formula:

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

In this case, the base is the x-intercept \( \frac{14}{3} \) and the height is the absolute value of the y-intercept \( |-14| = 14 \).

Substituting these values into the area formula:

\[ \text{Area} = \frac{1}{2} \times \frac{14}{3} \times 14 \]

Calculating the area:

\[ \text{Area} = \frac{1}{2} \times \frac{14 \times 14}{3} = \frac{1}{2} \times \frac{196}{3} = \frac{98}{3} \]

### Final Answer
The area of the triangle formed by the coordinate axes and the line \( y = 3x - 14 \) is \( \frac{98}{3} \).

---