What is the solution to the system?

-3x+4y=2z=-3 2x-4y-z=0 y=3x-13

It seems you have provided a system of equations, but the first equation isn't formatted clearly. Let's organize the system into a more readable form. I believe it might be:

\[
\begin{align*}
-3x + 4y + 2z &= -3 \\
2x - 4y - z &= 0 \\
y &= 3x - 13
\end{align*}
\]

This is a system of three equations in three variables: \(x\), \(y\), and \(z\).

I'll solve this system step by step:

1. **From the third equation** \( y = 3x - 13 \), we can substitute this value of \( y \) into the first and second equations to reduce the system to two equations in terms of \( x \) and \( z \).

2. Substitute \( y = 3x - 13 \) into the first two equations and solve for \( x \) and \( z \).

Let me perform these steps for you.from sympy import symbols, Eq, solve

# Define the variables
x, y, z = symbols('x y z')

# Given equations
eq1 = Eq(-3*x + 4*y + 2*z, -3)
eq2 = Eq(2*x - 4*y - z, 0)
eq3 = Eq(y, 3*x - 13)

# Substitute y = 3x - 13 into the first two equations
eq1_sub = eq1.subs(y, 3*x - 13)
eq2_sub = eq2.subs(y, 3*x - 13)

# Solve the system of equations for x and z
solution = solve([eq1_sub, eq2_sub], (x, z))

# Once x and z are found, solve for y
y_solution = eq3.subs(x, solution[x])

solution, y_solutionThe solution to the system of equations is:

\[
x = 5, \quad y = 2, \quad z = 2
\]

These values satisfy all three equations.

Would you like further details on the steps involved? Here are five related questions you could explore next:

1. What happens if the system had dependent equations leading to infinite solutions?
2. How can systems of nonlinear equations be solved?
3. How can this system be solved using matrices and determinants?
4. What are the graphical interpretations of systems with three variables?
5. How would the solution change if one equation was slightly altered?

**Tip:** When solving systems of equations, substitution is often an effective method, especially when one equation is already solved for a single variable.

Explore the solution to the system of equations -3x + 4y + 2z = -3, 2x - 4y - z = 0, y = 3x - 13. Detailed step-by-step method using substitution. Suitable for high school students.

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