Continuing from where you left off, let's simplify and reorganize these equations step by step. I'll rewrite the given system for clarity and solve it methodically.

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Given equations:

1. $-x_{11} + 3x_{21} = -x_{11} + x_{12}$
2. $-x_{12} + 3x_{22} = 3x_{11} + 2x_{12}$
3. $x_{11} + 2x_{21} = -x_{21} + x_{22}$
4. $x_{12} + 2x_{22} = 3x_{21} + 2x_{22}$

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Step 1: Simplify each equation

Equation 1:

$-x_{11} + 3x_{21} = -x_{11} + x_{12}$ Cancel $-x_{11}$ on both sides: $3x_{21} = x_{12}$

Equation 2:

$-x_{12} + 3x_{22} = 3x_{11} + 2x_{12}$ Combine like terms: $3x_{22} = 3x_{11} + 3x_{12}$ Divide through by 3: $x_{22} = x_{11} + x_{12}$

Equation 3:

$x_{11} + 2x_{21} = -x_{21} + x_{22}$ Combine like terms: $x_{11} + 3x_{21} = x_{22}$

Equation 4:

$x_{12} + 2x_{22} = 3x_{21} + 2x_{22}$ Cancel $2x_{22}$ on both sides: $x_{12} = 3x_{21}$

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Step 2: Substitute known relationships

From Equation 1 and Equation 4, we know: $x_{12} = 3x_{21}$ Substitute $x_{12} = 3x_{21}$ into Equation 2: $x_{22} = x_{11} + x_{12}$ $x_{22} = x_{11} + 3x_{21}$

Substitute $x_{22} = x_{11} + 3x_{21}$ into Equation 3: $x_{11} + 3x_{21} = x_{22}$ Since $x_{22} = x_{11} + 3x_{21}$, the equation is consistent.

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Final relationships:

• $x_{12} = 3x_{21}$
• $x_{22} = x_{11} + 3x_{21}$

This expresses $x_{12}$ and $x_{22}$ in terms of $x_{11}$ and $x_{21}$. Would you like to solve for numerical values or explore further simplifications?

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5 Related Questions:

1. How do these relationships affect the determinant of the system matrix?
2. Can we write this system in matrix form to analyze its rank?
3. What happens if specific numerical values are assigned to $x_{11}$ and $x_{21}$?
4. How would these equations behave under a parameterization of $x_{21}$?
5. Can we check for solutions using substitution and elimination explicitly?

Tip:

Always check for linear dependence when simplifying systems of linear equations, as it determines whether solutions are unique or parametric.