The area of the triangle formed by the line 5x - 3y + 15 = 0 with coordinate axes is

Options

A. 15 sq. units
B. 5 sq. units
C. 8 sq. units
D. `(15)/(2)` sq. units

Answer

To find the area of the triangle formed by the line \(5x - 3y + 15 = 0\) with the coordinate axes, we can follow these steps:

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

\[ 5x - 3(0) + 15 = 0 \implies 5x + 15 = 0 \implies 5x = -15 \implies x = -3 \]

So, the x-intercept is \((-3, 0)\).

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

\[ 5(0) - 3y + 15 = 0 \implies -3y + 15 = 0 \implies -3y = -15 \implies y = 5 \]

So, the y-intercept is \((0, 5)\).

### Step 3: Identify the vertices of the triangle
The triangle formed by the line and the coordinate axes has vertices at the origin \((0, 0)\), the x-intercept \((-3, 0)\), and the y-intercept \((0, 5)\).

### Step 4: Use the formula for the area of a triangle
The area \(A\) of a triangle formed by the vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by the formula:

\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Substituting the coordinates of the vertices:
- \((x_1, y_1) = (0, 0)\)
- \((x_2, y_2) = (-3, 0)\)
- \((x_3, y_3) = (0, 5)\)

### Step 5: Substitute the values into the area formula
\[ A = \frac{1}{2} \left| 0(0 - 5) + (-3)(5 - 0) + 0(0 - 0) \right| \]
\[ = \frac{1}{2} \left| 0 + (-3)(5) + 0 \right| \]
\[ = \frac{1}{2} \left| -15 \right| = \frac{1}{2} \times 15 = \frac{15}{2} \]

### Step 6: Final result
The area of the triangle is \(\frac{15}{2}\) square units.