x = 7a + 9b + 11c, y = 2b - 6a - 5c, z = 2c-3b+2a. Prove that, 2x - 3y + 5z = 42a - 3b + 47c.

Given the equations:

\[
x = 7a + 9b + 11c,
\]
\[
y = 2b - 6a - 5c,
\]
\[
z = 2c - 3b + 2a,
\]

we want to prove that:

\[
2x - 3y + 5z = 42a - 3b + 47c.
\]

### Step 1: Substitute the expressions for \(x\), \(y\), and \(z\)

First, substitute the given expressions for \(x\), \(y\), and \(z\) into the left-hand side of the equation:

\[
2x - 3y + 5z = 2(7a + 9b + 11c) - 3(2b - 6a - 5c) + 5(2c - 3b + 2a).
\]

### Step 2: Expand the expressions

Expand each term:

\[
2x = 2(7a + 9b + 11c) = 14a + 18b + 22c,
\]

\[
-3y = -3(2b - 6a - 5c) = -6b + 18a + 15c,
\]

\[
5z = 5(2c - 3b + 2a) = 10c - 15b + 10a.
\]

### Step 3: Combine the terms

Now, combine all these results:

\[
2x - 3y + 5z = (14a + 18b + 22c) + (18a - 6b + 15c) + (10a - 15b + 10c).
\]

Combine like terms:

\[
2x - 3y + 5z = (14a + 18a + 10a) + (18b - 6b - 15b) + (22c + 15c + 10c).
\]

Simplify further:

\[
2x - 3y + 5z = (42a) + (-3b) + (47c).
\]

### Conclusion

Since the right-hand side matches the expression given in the problem:

\[
2x - 3y + 5z = 42a - 3b + 47c,
\]

we have successfully proven that \(2x - 3y + 5z = 42a - 3b + 47c\).

Would you like more details or have any questions?

---

Here are 5 related questions:

1. How would the proof change if the equation for \(y\) was \(y = 2b + 6a - 5c\)?
2. Can you prove that \(x\), \(y\), and \(z\) form a linear combination?
3. What would the result be if you calculated \(3x + 2y - z\)?
4. How can this proof be applied to systems of linear equations in matrix form?
5. What would happen if we changed \(c\) to \(c + 1\) in all the equations?

**Tip:** When dealing with multiple variables, keeping your work organized by grouping like terms is crucial for simplifying and solving equations accurately.

Explore how to prove the linear combination x = 7a + 9b + 11c, y = 2b - 6a - 5c, z = 2c - 3b + 2a, leading to 2x - 3y + 5z = 42a - 3b + 47c. This problem involves advanced algebraic concepts and is suitable for advanced high school students.

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