The area of the region bounded by `y= x-5` with the coordinate axis is

Options

A. 25 sq units
B. 12.5 sq units
C. 50 sq units
D. 20 sq units

Answer

To find the area of the region bounded by the line \( y = x - 5 \) and the coordinate axes, we can follow these steps:

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

- **X-intercept**: Set \( y = 0 \)
\[ 0 = x - 5 \implies x = 5 \]
So, the x-intercept is \( (5, 0) \).

- **Y-intercept**: Set \( x = 0 \)
\[ y = 0 - 5 \implies y = -5 \]
So, the y-intercept is \( (0, -5) \).

### Step 2: Plot the points and the line
We can plot the points \( (5, 0) \) and \( (0, -5) \) on a coordinate plane. The line \( y = x - 5 \) will cross the x-axis at \( (5, 0) \) and the y-axis at \( (0, -5) \).

### Step 3: Identify the shape of the bounded region
The region bounded by the line and the coordinate axes forms a right triangle with vertices at \( (0, 0) \), \( (5, 0) \), and \( (0, -5) \).

### Step 4: 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 distance along the x-axis from \( (0, 0) \) to \( (5, 0) \), which is \( 5 \), and the height is the distance along the y-axis from \( (0, 0) \) to \( (0, -5) \), which is \( 5 \) (the absolute value since we consider area).

Substituting the values into the area formula:
\[ A = \frac{1}{2} \times 5 \times 5 = \frac{25}{2} = 12.5 \]

### Final Answer
The area of the region bounded by \( y = x - 5 \) with the coordinate axes is \( 12.5 \) square units.

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