https://file.aiquickdraw.com/mathbot/file/1733847112602fhr4cqy3.jpg

The given problem involves solving a system of linear algebraic equations (СЛАУ) using any method and providing the sum \(x + y + z + t\) as the answer. Here is the system extracted from the image:

\[
\begin{aligned}
1.\ & x - 2y + 3z + t = -1, \\
2.\ & 2x - 4y + 4z + 3t = -7, \\
3.\ & 5x - 6y + 2z + 3t = -17, \\
4.\ & -2x + 3y + z + 2t = 9.
\end{aligned}
\]

I'll solve this system step-by-step using Gaussian elimination. sympy import symbols, Eq, solve, Matrix

# Define variables
x, y, z, t = symbols('x y z t')

# Define the system of equations
equations = [
    Eq(x - 2*y + 3*z + t, -1),
    Eq(2*x - 4*y + 4*z + 3*t, -7),
    Eq(5*x - 6*y + 2*z + 3*t, -17),
    Eq(-2*x + 3*y + z + 2*t, 9)
]

# Convert to augmented matrix
augmented_matrix = Matrix([
    [1, -2, 3, 1, -1],
    [2, -4, 4, 3, -7],
    [5, -6, 2, 3, -17],
    [-2, 3, 1, 2, 9]
])

# Perform Gaussian elimination
reduced_matrix = augmented_matrix.rref()

# Extract solution from the reduced matrix
solution_matrix = solve(equations, (x, y, z, t))
solution_matrix, sum(solution_matrix.values())The solution to the system of equations is:

\[
x = 0, \, y = 3, \, z = 2, \, t = -1.
\]

The sum \(x + y + z + t\) is:

\[
0 + 3 + 2 - 1 = 4.
\]

Would you like to see the detailed steps or have any other questions? Here are some related questions to consider:

1. How can Gaussian elimination be applied step-by-step for such systems?
2. What other methods (e.g., substitution or matrix inversion) could solve this system?
3. How is the reduced row echelon form (RREF) determined during elimination?
4. What is the significance of consistency in a system of equations?
5. How can such problems be efficiently solved using computational tools?

**Tip:** When solving systems of linear equations, always check for special cases like infinite solutions or no solutions (inconsistent systems).

Solve the given system of linear algebraic equations (СЛАУ) by any method and provide the sum x + y + z + t.

Discover how to solve a system of four linear equations using Gaussian elimination. Includes step-by-step solutions, key concepts, and the sum x + y + z + t = 4. Suitable for Grades 10-12 and early university students.

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