https://file.aiquickdraw.com/mathbot/file/17255861512947v6u67mu.jpg

I see the uploaded image contains a list of mathematical problems. Let's tackle each one step-by-step. Here's the breakdown and solutions for the visible problems:

1. **Solve the following three-variable equation:**
   \[
   \begin{align*}
   3x - 5y + z &= 1 \\
   2x + y - 3z &= 2 \\
   x - 4y + 5z &= 3
   \end{align*}
   \]

2. **Solve the given quadratic equation by completing the square:**
   \[
   4x^2 - 12x + 5 = 0
   \]

3. **Write \( x^2-9 \) in terms of its factors.**
   \[
   x^2 - 9
   \]

4. **Write the function in terms of \( y \):**
   \[
   y = 2 \cos(x) + 5
   \]

5. **Factor the following trinomial completely:**
   \[
   6x^2 - 7x - 3 = 0
   \]

6. **Simplify the expression:**
   \[
   \frac{3x^2 - 3}{x^2 - 1} \div \frac{9x^2 - 12x + 4}{3x^2 + 5x - 2}
   \]

7. **Simplify the complex fraction:**
   \[
   \frac{\frac{3x-1}{4} + \frac{x+2}{3}}{\frac{x-2}{5} - \frac{x+3}{6}}
   \]

8. **Simplify the given expression using logarithmic identities:**
   \[
   \log_b x^a + \log_b y^a - \log_b z^a
   \]

Let's solve each in order starting from the first. I will begin with the system of linear equations:

**1. Solving the system of equations:**
\[
\begin{align*}
3x - 5y + z &= 1 \\
2x + y - 3z &= 2 \\
x - 4y + 5z &= 3
\end{align*}
\]

We can use methods like substitution or elimination. Let's solve it using matrix methods for clarity and efficiency.

We will set up the augmented matrix and perform row operations to find the values of \(x\), \(y\), and \(z\). Let's compute this.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to know anything else about these problems or any other mathematical concept? Here are a few related questions you might consider:

1. How do you solve a system of linear equations using the substitution method?
2. What is the method of completing the square in quadratic equations and why is it useful?
3. Can you explain how to factor expressions like \( x^2 - 9 \)?
4. How would you interpret the cosine function in the equation \( y = 2 \cos(x) + 5 \)?
5. What are logarithmic identities and how are they applied to simplify expressions like \( \log_b x^a + \log_b y^a - \log_b z^a \)?

**Tip:** When dealing with systems of equations, setting up an augmented matrix and using row reduction techniques can often provide a clear path to the solution.

Solve the following equations: [from image]

This page covers algebraic topics including solving quadratic equations, simplifying rational expressions, and working with logarithmic identities. Suitable for students in Grades 9-12.

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