If [2*p*+*q*5*r*−*s*​*p*−2*q*4*r*+3*s*​]=[411​−324​] then the value of *p*+*q*−*r*+2*s* is

Given the equation:

\[
[2p + q5r - sp - 2q4r + 3s] = [411 - 324]
\]

This equation can be simplified and broken down to find the values of \( p \), \( q \), \( r \), and \( s \).

Let's first simplify the left-hand side:

\[
2p + 5qr - sp - 8qr + 3s
\]

Now, let's solve the right-hand side:

\[
411 - 324 = 87
\]

So, the equation becomes:

\[
2p + 5qr - sp - 8qr + 3s = 87
\]

To solve for \( p + q - r + 2s \), we need the values of \( p \), \( q \), \( r \), and \( s \). Without specific values or additional equations, it's not possible to determine the individual values directly from this equation.

However, let's try expressing \( p + q - r + 2s \) in terms of \( 2p + 5qr - sp - 8qr + 3s = 87 \).

Unfortunately, we would need more information or specific values of \( p \), \( q \), \( r \), and \( s \) to solve this directly.

If you have any specific values or additional equations, please provide them to continue.

Would you like to explore this in more detail or clarify the problem further? Here are some related questions that could help you:

1. How do we simplify expressions involving multiple variables?
2. What are common techniques for solving systems of linear equations?
3. What does it mean to solve for a combination of variables like \( p + q - r + 2s \)?
4. How do algebraic expressions translate into geometric interpretations?
5. What role do coefficients play in determining the values of variables?

**Tip:** When solving equations involving multiple variables, try to isolate terms or seek additional equations to find each variable's value.

Explore the solution to a complex linear equation involving multiple variables. Understand the process of simplifying expressions and solving systems of equations. Suitable for advanced high school students.

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