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Suppose  Consider the following system of linear equations. x + y + z = α, x + βy + z = γ, x + y + αz = β. If this system has at least one solution, then which of the following statements is (are) true? a)If α = 1 then γ = 1 b)If β = 1 then γ = α c)If β  ≠ 1 then α = 1 d)If γ = 1 then α = 1?
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Suppose  Consider the following system of linear equations. x + y + z ...
Analysis of the System of Equations
Given the system of linear equations:
1. \(x + y + z = \alpha\) (Equation 1)
2. \(x + \beta y + z = \gamma\) (Equation 2)
3. \(x + y + \alpha z = \beta\) (Equation 3)
To determine the conditions under which this system has at least one solution, we can analyze the relationships between the variables and the constants \( \alpha \), \( \beta \), and \( \gamma \).
Condition Analysis
- If \( \alpha = 1 \) then \( \gamma = 1 \)
- Substituting \( \alpha = 1 \) into Equations 1 and 3 shows that all equations align under specific conditions, particularly leading to \( \gamma = 1 \) when evaluated appropriately.
- True
- If \( \beta = 1 \) then \( \gamma = \alpha \)
- When substituting \( \beta = 1 \) into Equations 2 and 3, we observe that \( \gamma\) becomes directly linked to \( \alpha\), confirming this relationship.
- True
- If \( \beta \neq 1 \) then \( \alpha = 1 \)
- This condition does not necessarily hold. The system can have solutions for varying \( \alpha\) values when \( \beta\) is not equal to 1.
- False
- If \( \gamma = 1 \) then \( \alpha = 1\)
- The condition does not guarantee \( \alpha = 1\) since \( \gamma\) can equal 1 under different values of \( \alpha\).
- False
Conclusion
The statements that are true are:
- a) If \( \alpha = 1 \) then \( \gamma = 1 \)
- b) If \( \beta = 1 \) then \( \gamma = \alpha \)
The other two statements are false.
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Suppose  Consider the following system of linear equations. x + y + z = α, x + βy + z = γ, x + y + αz = β. If this system has at least one solution, then which of the following statements is (are) true? a)If α = 1 then γ = 1 b)If β = 1 then γ = α c)If β  ≠ 1 then α = 1 d)If γ = 1 then α = 1?
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