Questions: Solve the system of equations. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. 3x = 2y - 4z - 13 -3(x + y) = -4y + z - 1 7(x - y) + z = 3x - 6y + 4 The system has one solution. The solution set is .

Solve the system of equations. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. 3x = 2y - 4z - 13 -3(x + y) = -4y + z - 1 7(x - y) + z = 3x - 6y + 4 The system has one solution. The solution set is .
Transcript text: Solve the system of equations. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. \[ \begin{array}{l} 3 x=2 y-4 z-13 \\ -3(x+y)=-4 y+z-1 \\ 7(x-y)+z=3 x-6 y+4 \end{array} \] The system has one solution. The solution set is $\{$ $\square$ $\square$ $\square$ $\}$.
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Solution

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Solution Steps

To solve the given system of equations, we can use matrix methods such as Gaussian elimination or matrix inversion if the system is consistent and has a unique solution. We will first represent the system in matrix form and then use a suitable method to find the solution. If the system is inconsistent or the equations are dependent, we will determine the nature of the solutions.

Step 1: Represent the System of Equations

The given system of equations can be represented in matrix form as follows:

\[ \begin{align_} 3x - 2y + 4z &= -13 \\ -3x - 3y + z &= 1 \\ 7x - 7y + z &= 4 \end{align_} \]

Step 2: Solve the System

Upon solving the system, we find that there is one unique solution given by:

\[ \begin{align_} x &\approx 0.6316 \\ y &\approx 0.5526 \\ z &\approx -3.4474 \end{align_} \]

Step 3: Write the Solution Set

The solution set can be expressed as:

\[ \{ x, y, z \} = \{ 0.6316, 0.5526, -3.4474 \} \]

Final Answer

The system has one unique solution. The solution set is

\[ \boxed{\{ 0.6316, 0.5526, -3.4474 \}} \]

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