Find area parallelogram lines `y=mx, y=mx+1, y=nx and y=nx+1` equal to:

Answer

To find the area of the parallelogram formed by the lines \( y = mx \), \( y = mx + 1 \), \( y = nx \), and \( y = nx + 1 \), we can follow these steps:

### Step 1: Identify the lines and their intersections
The lines given are:
1. \( y = mx \)
2. \( y = mx + 1 \)
3. \( y = nx \)
4. \( y = nx + 1 \)

We need to find the points of intersection of these lines to determine the vertices of the parallelogram.

### Step 2: Find the intersection points
To find the intersection of the lines \( y = mx \) and \( y = nx + 1 \):
Set \( mx = nx + 1 \).
Rearranging gives:
\[ mx - nx = 1 \implies (m - n)x = 1 \implies x = \frac{1}{m - n} \]
Substituting \( x \) back into \( y = mx \):
\[ y = m\left(\frac{1}{m - n}\right) = \frac{m}{m - n} \]
Thus, one vertex \( A \) is \( \left(\frac{1}{m - n}, \frac{m}{m - n}\right) \).

Next, find the intersection of the lines \( y = mx + 1 \) and \( y = nx \):
Set \( mx + 1 = nx \).
Rearranging gives:
\[ mx - nx = -1 \implies (m - n)x = -1 \implies x = \frac{-1}{m - n} \]
Substituting \( x \) back into \( y = mx + 1 \):
\[ y = m\left(\frac{-1}{m - n}\right) + 1 = \frac{-m}{m - n} + 1 = \frac{-m + (m - n)}{m - n} = \frac{n}{m - n} \]
Thus, another vertex \( B \) is \( \left(\frac{-1}{m - n}, \frac{n}{m - n}\right) \).

### Step 3: Calculate the area of the parallelogram
The area \( A \) of the parallelogram can be calculated using the formula:
\[ A = \text{base} \times \text{height} \]
Here, the base can be taken as the distance between the lines \( y = mx \) and \( y = mx + 1 \), which is \( 1 \). The height is the distance between the lines \( y = nx \) and \( y = mx \).

The height can be calculated as:
\[ \text{Height} = \frac{1}{|m - n|} \]
Thus, the area of the parallelogram is:
\[ A = \text{base} \times \text{height} = 1 \times \frac{1}{|m - n|} = \frac{1}{|m - n|} \]

### Final Answer
The area of the parallelogram is:
\[ \text{Area} = \frac{1}{|m - n|} \]