The system of linear equations in x1, x2, x3 where α, β ∈ ℝ has...
Understand the Problem
The question presents a system of linear equations involving the variables x1, x2, and x3 and parameters α and β. It seeks to analyze this system to understand its properties or solutions based on the values of α and β.
Answer
A unique solution if $ \alpha \neq -4 $, infinitely many solutions if $ \alpha = -4 $ and $ \beta = 4 $, and no solutions if $ \alpha = -4 $ and $ \beta \neq 4 $.
Answer for screen readers
The system has:
- A unique solution if ( \alpha \neq -4 ).
- Infinitely many solutions if ( \alpha = -4 ) and ( \beta = 4 ).
- No solutions if ( \alpha = -4 ) and ( \beta \neq 4 ).
Steps to Solve
- Write the System in Matrix Form
The given system of equations is represented as:
$$
\begin{pmatrix}
1 & 1 & 0 \
0 & -1 & 2 \
2 & 0 & \alpha
\end{pmatrix}
\begin{pmatrix}
x_1 \
x_2 \
x_3
\end{pmatrix}
\begin{pmatrix} 3 \ 1 \ \beta \end{pmatrix} $$
- Set Up the Augmented Matrix
Create the augmented matrix that includes the coefficients and constants: $$ \left( \begin{array}{ccc|c} 1 & 1 & 0 & 3 \ 0 & -1 & 2 & 1 \ 2 & 0 & \alpha & \beta \end{array} \right) $$
- Perform Row Operations for Row Echelon Form
Start applying row operations to simplify the augmented matrix. From the augmented matrix, subtract 2 times the first row from the third row:
- Row 3: ( \text{Row}_3 - 2 \times \text{Row}_1 )
Resulting in: $$ \left( \begin{array}{ccc|c} 1 & 1 & 0 & 3 \ 0 & -1 & 2 & 1 \ 0 & -2 & \alpha & \beta - 6 \end{array} \right) $$
- Continue Row Operations
Next, simplify the second row to make a leading 1:
- Row 2: ( -1 \times \text{Row}_2 )
And then add 2 times the second row to the third row:
- Row 3: ( \text{Row}_3 + 2 \times \text{Row}_2 )
Resulting in: $$ \left( \begin{array}{ccc|c} 1 & 1 & 0 & 3 \ 0 & 1 & -2 & -1 \ 0 & 0 & \alpha + 4 & \beta - 4 \end{array} \right) $$
- Analyze Solutions Based on Parameters α and β
The system has different cases based on the value of ( \alpha + 4 ):
- If ( \alpha + 4 \neq 0 ): The system has a unique solution.
- If ( \alpha + 4 = 0 ): Analyze the condition:
- If ( \beta - 4 = 0 ): Infinitely many solutions (dependent).
- If ( \beta - 4 \neq 0 ): No solution (inconsistent).
The system has:
- A unique solution if ( \alpha \neq -4 ).
- Infinitely many solutions if ( \alpha = -4 ) and ( \beta = 4 ).
- No solutions if ( \alpha = -4 ) and ( \beta \neq 4 ).
More Information
This analysis reveals the dependence of the system's solutions on the parameters α and β. Understanding these relationships is crucial in linear algebra for solving systems of equations effectively.
Tips
- Failing to simplify the augmented matrix correctly during row operations.
- Confusing conditions for unique, infinite, or no solutions.
- Not recognizing that both α and β play a crucial role in determining the nature of the solution.
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