https://file.aiquickdraw.com/mathbot/file/1728532249864azjk8m2a.jpg

Let's solve the two equations visible in the image.

**Equation 6:**

\[
\frac{1}{x} - \frac{x}{6} = \frac{2}{3}
\]

To solve this equation, we need to find a common denominator.

1. The common denominator for \(\frac{1}{x}\) and \(\frac{x}{6}\) is \(6x\).
2. Rewrite the equation:
   \[
   \frac{6}{6x} - \frac{x^2}{6x} = \frac{2}{3}
   \]
   This simplifies to:
   \[
   \frac{6 - x^2}{6x} = \frac{2}{3}
   \]
3. Cross-multiply:
   \[
   3(6 - x^2) = 12x
   \]
   Expanding the equation:
   \[
   18 - 3x^2 = 12x
   \]
4. Rearrange all terms to one side:
   \[
   3x^2 + 12x - 18 = 0
   \]
5. Divide by 3:
   \[
   x^2 + 4x - 6 = 0
   \]
6. Solve using the quadratic formula:
   \[
   x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-6)}}{2(1)}
   \]
   \[
   x = \frac{-4 \pm \sqrt{16 + 24}}{2}
   \]
   \[
   x = \frac{-4 \pm \sqrt{40}}{2}
   \]
   \[
   x = \frac{-4 \pm 2\sqrt{10}}{2}
   \]
   \[
   x = -2 \pm \sqrt{10}
   \]

Thus, the solutions are:

\[
x = -2 + \sqrt{10} \quad \text{or} \quad x = -2 - \sqrt{10}
\]

---

**Equation 7:**

\[
\frac{4}{t - 3} + \frac{t}{2} = -2
\]

To solve this:

1. Multiply through by \(2(t - 3)\) to eliminate the denominators:
   \[
   2(t - 3) \left( \frac{4}{t - 3} + \frac{t}{2} \right) = 2(t - 3)(-2)
   \]
   This simplifies to:
   \[
   8 + t(t - 3) = -4(t - 3)
   \]
2. Expand both sides:
   \[
   8 + t^2 - 3t = -4t + 12
   \]
   Simplify:
   \[
   t^2 + t - 4 = 0
   \]
3. Solve this quadratic equation using the quadratic formula:
   \[
   t = \frac{-1 \pm \sqrt{1^2 - 4(1)(-4)}}{2(1)}
   \]
   \[
   t = \frac{-1 \pm \sqrt{1 + 16}}{2}
   \]
   \[
   t = \frac{-1 \pm \sqrt{17}}{2}
   \]

Thus, the solutions are:

\[
t = \frac{-1 + \sqrt{17}}{2} \quad \text{or} \quad t = \frac{-1 - \sqrt{17}}{2}
\]

---

Do you want more details or have any questions on these solutions? Here are some related questions for further exploration:

1. Can you find the approximate numerical values of \(x\) for both solutions of equation 6?
2. How would you check if your solutions for \(x\) and \(t\) are correct by substitution?
3. What happens if you change the constant terms in both equations? How does it affect the solutions?
4. Can you graph the quadratic equations obtained in both problems? What do the graphs reveal about the solutions?
5. How would you solve these problems using a different method, such as completing the square?

**Tip:** Always simplify the equation step-by-step to avoid mistakes when dealing with fractions and quadratics.

Solve the two equations: (1/x) - (x/6) = 2/3 and (4/(t-3)) + (t/2) = -2

Learn how to solve two challenging equations involving fractions and quadratic terms: (1/x) - (x/6) = 2/3 and (4/(t-3)) + (t/2) = -2. This step-by-step solution covers key algebraic concepts, such as manipulating fractions, cross-multiplying, and solving quadratic equations using the quadratic formula.

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