If {x^3} + 1 + 6xy = 8{y^3} and x \ne 1, then the value of 2y – x is:

1

2

–1

None of these

To find the value of \(2y - x\) given the equation:

\[
x^3 + 1 + 6xy = 8y^3
\]

where \(x \neq 1\), let's proceed step-by-step.

### Step 1: Rewrite the Equation

The given equation is:

\[
x^3 + 1 + 6xy = 8y^3
\]

### Step 2: Rearrange the Equation

Rearrange the equation to isolate all terms involving \(x\) and \(y\):

\[
x^3 + 6xy + 1 = 8y^3
\]

### Step 3: Check Possible Values

To find a relation between \(x\) and \(y\), let's consider if there is a simple solution by substituting small integers for \(x\) and \(y\).

Let's substitute \(x = 2y\) in the equation to test if it simplifies:

\[
x = 2y \implies (2y)^3 + 6(2y)y + 1 = 8y^3
\]

Substitute and simplify:

\[
8y^3 + 12y^2 + 1 = 8y^3
\]

Subtract \(8y^3\) from both sides:

\[
12y^2 + 1 = 0
\]

This equation doesn't yield a valid real value for \(y\), so we will test another approach.

### Step 4: Substitute \(2y - x = 1\)

Suppose \(2y - x = 1\). Then we can express \(x\) in terms of \(y\):

\[
x = 2y - 1
\]

Substitute this into the original equation:

\[
(2y - 1)^3 + 6(2y - 1)y + 1 = 8y^3
\]

### Step 5: Simplify the Substituted Expression

Expand \((2y - 1)^3\):

\[
(2y - 1)^3 = 8y^3 - 12y^2 + 6y - 1
\]

Now substitute back:

\[
8y^3 - 12y^2 + 6y - 1 + 6(2y^2 - y) + 1 = 8y^3
\]

Simplify:

\[
8y^3 - 12y^2 + 6y - 1 + 12y^2 - 6y + 1 = 8y^3
\]

Combine like terms:

\[
8y^3 + 0y^2 + 0y + 0 = 8y^3
\]

The equation simplifies to \(8y^3 = 8y^3\), which is always true.

### Conclusion

This means our assumption \(2y - x = 1\) is correct. Therefore, the value of \(2y - x\) is:

\[
\boxed{1}
\]

Would you like more details or have any questions?

### Related Questions

1. How do we solve polynomial equations with multiple variables?
2. Can we find the value of \(x\) and \(y\) given additional constraints?
3. How do we graphically interpret the equation \(x^3 + 1 + 6xy = 8y^3\)?
4. What are the methods for verifying the validity of a proposed solution?
5. How can we simplify polynomial expressions effectively?

**Tip:** Always check your assumptions by substituting back into the original equation to verify the solution.

Explore the solution to the mathematical equation x^3 + 1 + 6xy = 8y^3 step-by-step. Understand the algebraic manipulation involved and the verification of assumptions. Suitable for advanced high school students.

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