Solving for x When fx 2x^7 and fx -1

When dealing with algebraic equations in the context of mathematical problem-solving, it is essential to understand the processes involved in solving equations. This article will walk through an example of solving for x when fx 2x^7 and fx -1.

Introduction

When two functions, fx 2x^7 and fx -1, are given, we can use the substitution property of equality to solve for x. This process involves substituting one equation into the other and solving for the variable.

Step-by-Step Solution

Equation 1: fx 2x^7

Equation 2: fx -1

To solve for x, we start by setting the two equations equal to each other:

2x^7 -1

Following the steps to solve for x through algebraic manipulation:

2x^7 -1

Subtract 7 from both sides:

2x^7 - 7 -1 - 7

Simplify:

2x^7 -8

Divide both sides by 2:

x^7 -4

Take the seventh root of both sides:

x -4

To verify the solution, we substitute x -4 back into the original equations.

Verification

fx 2x^7

f(-4) 2(-4)^7

f(-4) 2(-16384)

f(-4) -32768

This does not equal -1, indicating a mistake in the interpretation or the steps employed. Let's correct this using the fx -1 equation directly:

2x^7 -1

Subtract 7 does not simplify further:

2x^7 -1

Divide by 2:

x^7 -1/2

Take the seventh root:

x (-1/2)^(1/7)

This does not yield a simple integer solution, indicating a different approach is needed. Revisiting the substitution:

2x^7 -1

Divide by 2:

x^7 -1/2

Take the seventh root:

x -1

Conclusion

Upon re-evaluation:

2(-1)^7 -1

Therefore, the correct answer is:

x -1

These steps demonstrate the importance of careful algebraic manipulation and verification in solving equations. The correct solution to the given problem is x -1.

Understanding these methods is crucial for tackling more complex algebraic problems in both academic and practical settings.