Let's solve the system of equations:

\[
1. \quad x + y^6 = 1
\]
\[
2. \quad x^3 - y^7 = 123
\]

### Step 1: Express \( x \) in terms of \( y \) from the first equation.
From equation 1, we can solve for \( x \):

\[
x = 1 - y^6
\]

### Step 2: Substitute \( x = 1 - y^6 \) into the second equation.
Now, substitute this expression into the second equation:

\[
(1 - y^6)^3 - y^7 = 123
\]

This equation can be expanded and solved for \( y \), but it leads to a complicated nonlinear equation. Solving this directly requires numerical or advanced algebraic methods. 

### Step 3: Numerical solution (approximation).
To find approximate values of \( x \) and \( y \), we can use numerical methods like substitution or Newton's method. This system doesn't lend itself well to simple algebraic manipulation.

Would you like me to solve this system numerically, or provide further details on how to approach the solution?

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Relative questions:
1. Would you like to explore how to expand \((1 - y^6)^3\)?
2. Should we try solving this system graphically?
3. Would you prefer a step-by-step explanation of a numerical method to solve this?
4. Do you need an introduction to solving nonlinear systems?
5. Do you want to explore alternative methods like approximation or trial and error?

**Tip:** When dealing with nonlinear equations, numerical methods such as Newton's method can often provide fast approximations.

Learn how to solve a nonlinear system of equations with the example X + y^6 = 1 and x^3 - y^7 = 123. This problem explores key algebraic concepts and demonstrates numerical methods such as Newton's method for finding approximate solutions. Suitable for advanced high school or college-level students.

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