The given equation is:

\[
10 - 5q = \frac{1}{3q}
\]

To solve this equation, let's follow the steps:

### Step 1: Multiply both sides by \(3q\)
This will eliminate the fraction on the right-hand side.

\[
(10 - 5q) \cdot 3q = 1
\]

Expanding the left-hand side:

\[
30q - 15q^2 = 1
\]

### Step 2: Rearrange the equation
Bring all terms to one side to form a quadratic equation:

\[
15q^2 - 30q + 1 = 0
\]

### Step 3: Solve the quadratic equation
Use the quadratic formula to solve for \(q\):

\[
q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(15q^2 - 30q + 1 = 0\), the coefficients are:
- \(a = 15\)
- \(b = -30\)
- \(c = 1\)

Substitute the values into the quadratic formula:

\[
q = \frac{-(-30) \pm \sqrt{(-30)^2 - 4(15)(1)}}{2(15)}
\]

Simplifying:

\[
q = \frac{30 \pm \sqrt{900 - 60}}{30}
\]
\[
q = \frac{30 \pm \sqrt{840}}{30}
\]
\[
q = \frac{30 \pm 28.98}{30}
\]

### Step 4: Calculate the solutions

\[
q_1 = \frac{30 + 28.98}{30} = \frac{58.98}{30} \approx 1.966
\]
\[
q_2 = \frac{30 - 28.98}{30} = \frac{1.02}{30} \approx 0.034
\]

### Final solutions:

\[
q \approx 1.97 \quad \text{or} \quad q \approx 0.034
\]

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. How do you solve quadratic equations using the quadratic formula?
2. What are the steps to eliminate fractions in an equation?
3. How do you expand expressions involving parentheses?
4. How do you simplify square roots for non-perfect squares?
5. What is the discriminant and how does it affect the number of solutions?

**Tip:** Always check if you can factor the quadratic equation before applying the quadratic formula for a faster solution!

Explore how to solve the equation 10 - 5q = 1 / (3q) step by step. This solution covers eliminating fractions, expanding expressions, and applying the quadratic formula to find the roots. Suitable for Grades 10-12 students studying quadratic equations and algebra.

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