If 5x+4y=54 and 4x+5y=45 then find the value of (x+y)  (x-y)

To solve the system of equations and find the value of \((x + y)(x - y)\), follow these steps:

1. **Write down the given system of equations:**

\[
\begin{cases}
5x + 4y = 54 \quad \text{(Equation 1)} \\
4x + 5y = 45 \quad \text{(Equation 2)}
\end{cases}
\]

2. **Solve the system using the elimination method:**

First, multiply Equation 1 by 5 and Equation 2 by 4 to align the coefficients for elimination:

\[
\begin{cases}
25x + 20y = 270 \quad \text{(Equation 3)} \\
16x + 20y = 180 \quad \text{(Equation 4)}
\end{cases}
\]

3. **Subtract Equation 4 from Equation 3 to eliminate \(y\):**

\[
(25x + 20y) - (16x + 20y) = 270 - 180
\]

\[
9x = 90
\]

4. **Solve for \(x\):**

\[
x = \frac{90}{9} = 10
\]

5. **Substitute \(x\) back into one of the original equations to find \(y\):**

Using Equation 1:

\[
5(10) + 4y = 54
\]

\[
50 + 4y = 54
\]

\[
4y = 54 - 50
\]

\[
4y = 4
\]

\[
y = \frac{4}{4} = 1
\]

6. **Now that we have \(x = 10\) and \(y = 1\), we can find \((x + y)(x - y)\):**

\[
(x + y)(x - y) = (10 + 1)(10 - 1) = 11 \times 9 = 99
\]

Therefore, the value of \((x + y)(x - y)\) is \(\boxed{99}\).

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How do you solve a system of linear equations using substitution?
2. What are the other methods to solve a system of linear equations?
3. Can you explain the geometric interpretation of solving a system of linear equations?
4. How does the determinant of the coefficient matrix relate to the solution of the system?
5. How can we use matrices to solve a system of linear equations?
6. What is the role of the elimination method in solving linear systems?
7. How can you check if a system of equations has no solution, one solution, or infinitely many solutions?
8. How do linear equations relate to real-world problems?

**Tip:** Always double-check your solutions by substituting the values back into the original equations to ensure they satisfy both equations.

Learn how to solve a system of linear equations to find the value of (x + y)(x - y). This problem involves two linear equations with two variables, requiring the elimination method to find the values of x and y. Suitable for students in Grades 9-12.

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