Let's solve this system of equations step by step.

We have two equations:

1. \[
\frac{2}{7} \cdot 3x = 9x + 1
\]
2. \[
\frac{27}{3x} = 9x + 1
\]

---

### First Equation:

\[
\frac{2}{7} \cdot 3x = 9x + 1
\]

First, simplify the left-hand side:
\[
\frac{6x}{7} = 9x + 1
\]

Now, multiply both sides by 7 to eliminate the denominator:
\[
6x = 63x + 7
\]

Subtract \(63x\) from both sides:
\[
6x - 63x = 7
\]
\[
-57x = 7
\]

Solve for \(x\):
\[
x = \frac{7}{-57} = -\frac{1}{9}
\]

---

### Second Equation:

\[
\frac{27}{3x} = 9x + 1
\]

Multiply both sides by \(3x\) to eliminate the denominator:
\[
27 = (9x + 1)(3x)
\]

Expand the right-hand side:
\[
27 = 27x^2 + 3x
\]

Move everything to one side:
\[
27x^2 + 3x - 27 = 0
\]

Now, divide through by 3 to simplify:
\[
9x^2 + x - 9 = 0
\]

Solve this quadratic equation using the quadratic formula:
\[
x = \frac{-1 \pm \sqrt{1^2 - 4(9)(-9)}}{2(9)}
\]
\[
x = \frac{-1 \pm \sqrt{1 + 324}}{18}
\]
\[
x = \frac{-1 \pm \sqrt{325}}{18}
\]
\[
x = \frac{-1 \pm 18.03}{18}
\]

So, the two possible solutions are:
\[
x_1 = \frac{-1 + 18.03}{18} \approx 0.945
\]
\[
x_2 = \frac{-1 - 18.03}{18} \approx -1.056
\]

---

### Final Answer:
- From the first equation: \(x = -\frac{1}{9}\)
- From the second equation: \(x_1 \approx 0.945\), \(x_2 \approx -1.056\)

Do you have any questions or would you like more details on the steps?

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Here are 5 related questions:
1. What happens if we change the constant in the first equation?
2. How do you derive the quadratic formula?
3. What are some real-world applications of solving quadratic equations?
4. How can you solve equations involving fractions more efficiently?
5. How do you graph the equation \(9x^2 + x - 9 = 0\)?

**Tip:** When dealing with fractions in equations, always try to eliminate the denominator early by multiplying both sides by the least common denominator (LCD).

Discover how to solve fractional and quadratic equations such as 2/7 * 3x = 9x + 1 and 27 / 3x = 9x + 1. This detailed explanation covers key algebraic concepts, including solving linear equations, applying the quadratic formula, and working with fractions. Ideal for students in Grades 8-10.

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2 7 3 x

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