https://file.aiquickdraw.com/mathbot/file/17260564976346y66v4fd.jpg

Let's break down the information provided in the problem and formulate the required linear equations:

### Given:
- Let the original dimensions of the plot be \( x \) (length) and \( y \) (breadth).
- **First condition:** If the length is decreased by 25 m and the breadth is increased by 25 m, the area increases by 625 m².
  \[
  (x - 25)(y + 25) = xy + 625
  \]
  Expanding:
  \[
  xy + 25x - 25y - 625 = xy + 625
  \]
  Simplifying:
  \[
  25x - 25y = 1250 \quad \Rightarrow \quad x - y = 50 \quad \text{(Equation 1)}
  \]

- **Second condition:** If the length is decreased by 20 m and the breadth is increased by 10 m, the area decreases by 200 m².
  \[
  (x - 20)(y + 10) = xy - 200
  \]
  Expanding:
  \[
  xy + 10x - 20y - 200 = xy - 200
  \]
  Simplifying:
  \[
  10x - 20y = 0 \quad \Rightarrow \quad x = 2y \quad \text{(Equation 2)}
  \]

### Part (a): Linear equations
From the given conditions, the two linear equations are:
1. \( x - y = 50 \)
2. \( x = 2y \)

### Part (b): Find the dimensions of the plot using matrix method

To solve using matrices, rewrite the system of equations:
1. \( x - y = 50 \)
2. \( x - 2y = 0 \)

In matrix form, this becomes:
\[
\begin{pmatrix} 
1 & -1 \\
1 & -2 
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
50 \\
0
\end{pmatrix}
\]

We will solve this system using the inverse of the coefficient matrix. Let me compute this for you.The dimensions of the plot of land are:
- Length \( x = 100 \, \text{m} \)
- Breadth \( y = 50 \, \text{m} \)

Would you like further details or have any questions?

Here are some related questions to explore:
1. What is the total area of the original plot?
2. How would the area change if both dimensions were increased by 10%?
3. Can you verify the solutions using substitution into the original conditions?
4. How would the system change if the area increased by a different amount in the first condition?
5. How can we represent such problems geometrically?

**Tip:** Matrix methods, especially solving linear systems, are efficient for multiple equations, even for larger systems.

An architect is developing a plot of land for a commercial complex. When asked about the dimensions of the plot, he said that if the length is decreased by 25 m and the breadth is increased by 25 m, then its area increases by 625 m². If the length is decreased by 20 m and the breadth is increased by 10 m, then its area decreases by 200 m².

Learn how to solve a system of linear equations to find the dimensions of a plot of land. This problem involves using algebra and matrix methods, based on changes in area when length and breadth are adjusted. Suitable for students in Grades 10-12.

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