The given problem requires solving a system of three linear equations for $x$, $y$, and $z$. Here's the system extracted from the image:

$\text{1. } x + y + 2z = 9$ $\text{2. } 2x + 4y - 3z = 1$ $\text{3. } 3x + 6y - 5z = 0$

Let's solve this step by step:

Step 1: Write equations in standard form

The equations are already written in standard form:

1. $x + y + 2z = 9$
2. $2x + 4y - 3z = 1$
3. $3x + 6y - 5z = 0$

Step 2: Eliminate one variable

We can eliminate one variable, say $x$, by combining the equations. Let me solve this system. sympy import symbols, Eq, solve

Define variables

x, y, z = symbols('x y z')

Define the equations

eq1 = Eq(x + y + 2z, 9) eq2 = Eq(2x + 4y - 3z, 1) eq3 = Eq(3x + 6y - 5*z, 0)

Solve the system of equations

solution = solve((eq1, eq2, eq3), (x, y, z)) solutionThe solution to the system of equations is:

$x = 1, \quad y = 2, \quad z = 3$

Would you like a detailed explanation of the steps, or do you have further questions? Here are some related questions you might explore:

1. How do you eliminate variables step-by-step using substitution or elimination?
2. What is the geometric interpretation of solving a system of three equations in three variables?
3. Can these equations be solved using matrix methods like Gaussian elimination?
4. How does this system behave if one equation is altered slightly?
5. What are the conditions for a system of linear equations to have no solution or infinitely many solutions?

Tip: Verify your solution by substituting $x = 1$, $y = 2$, and $z = 3$ back into the original equations!