((700*x+30)/(6-100*x)-7)^(2)+0.5*((250*x+20)/(6-50*x)-5)^(2)=1

To solve the given equation:

\[
\left( \frac{700x + 30}{6 - 100x} - 7 \right)^2 + 0.5 \left( \frac{250x + 20}{6 - 50x} - 5 \right)^2 = 1,
\]

we will approach this step-by-step:

### Step 1: Simplify Each Term
Let's simplify each fraction inside the parentheses:

1. **First Term:**

\[
\frac{700x + 30}{6 - 100x} - 7
\]

Combine the expression into a single fraction:

\[
\frac{700x + 30 - 7(6 - 100x)}{6 - 100x} = \frac{700x + 30 - 42 + 700x}{6 - 100x} = \frac{1400x - 12}{6 - 100x}
\]

2. **Second Term:**

\[
\frac{250x + 20}{6 - 50x} - 5
\]

Combine this expression into a single fraction:

\[
\frac{250x + 20 - 5(6 - 50x)}{6 - 50x} = \frac{250x + 20 - 30 + 250x}{6 - 50x} = \frac{500x - 10}{6 - 50x}
\]

### Step 2: Substitute Simplified Terms into the Equation

Substitute the simplified expressions back into the original equation:

\[
\left( \frac{1400x - 12}{6 - 100x} \right)^2 + 0.5 \left( \frac{500x - 10}{6 - 50x} \right)^2 = 1
\]

### Step 3: Solve the Equation

To solve this equation, consider a substitution for each fraction, let:

\[
y = \frac{1400x - 12}{6 - 100x}, \quad z = \frac{500x - 10}{6 - 50x}
\]

Now, the equation becomes:

\[
y^2 + 0.5z^2 = 1
\]

Using the definitions of \(y\) and \(z\), solve for \(x\) in each case:

\[
y(6 - 100x) = 1400x - 12 \implies 6y - 100xy = 1400x - 12 \implies 6y + 12 = 1400x + 100xy \implies  x = \frac{6y + 12}{1400 + 100y}
\]

Similarly, solve for \(x\) in terms of \(z\):

\[
z(6 - 50x) = 500x - 10 \implies 6z - 50xz = 500x - 10 \implies 6z + 10 = 500x + 50xz \implies x = \frac{6z + 10}{500 + 50z}
\]

Plug these into the original equation and solve the quadratic equations obtained for \(y\) and \(z\) numerically or symbolically. This is a complicated nonlinear equation and generally requires numerical methods or specific software (like MATLAB or Python's SymPy) for an exact solution.

### Step 4: Conclusion

The solution for \(x\) involves solving these equations numerically:

\[
y^2 + 0.5z^2 = 1
\]

Substitute back to find specific \(x\) values for which the equation holds.

If you need more detailed steps or have any questions, let me know!

### Further Exploration

1. What are the steps to solve a nonlinear equation numerically?
2. How can we simplify equations using algebraic manipulation?
3. How does substitution work in simplifying complex equations?
4. What are the different methods to solve quadratic equations?
5. How do graphical methods help solve equations like these?

**Tip:** Always double-check your algebra when simplifying complex fractions!

Learn how to solve the nonlinear equation ((700*x+30)/(6-100*x)-7)^(2)+0.5*((250*x+20)/(6-50*x)-5)^(2)=1 with a detailed step-by-step solution. This problem covers key concepts in nonlinear equations, fraction simplification, and substitution methods. Suitable for advanced students in Grades 11-12.

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