How would you solve the following system of linear equations: 2x_1 + (3+a)x_2 + 2x_3 = 2+aZ

Question

How would you solve the following system of linear equations:

2 x_{1} + (3 + a) x_{2} + 2 x_{3} = 2 + a

x_{1} + a x_{2} + 2 x_{3} = a

a x_{1} + 2 x_{2} + 2 a x_{3} = 0

assuming that

a \neq \pm \sqrt{2}

? I feel confident in solving linear equation systems with just constants as the coefficients but the variable coefficient a is what gives me problems.
Here is the solution I find: solution as augmented matrix but how would the assumption about a make the reduced echelon form any different compared to an assumption about e.g.

a \neq \pm \sqrt{2}

?

Answer 1

With

x_{1} = a - a x_{2} - 2 x_{3}

can we eliminate

x_{1}

:

x_{2} (3 - a) - 2 x_{3} = 2 - a

x_{2} (2 - a^{2}) + 4 a x_{3} = - a^{2}

And then using

x_{3} = \frac{x_{2} (3 - a) - (2 - a)}{2}

to eliminate

x_{3}

. Then you will get

x_{2} (2 - a^{2}) + 2 a (3 - a) x_{2} - 2 a (2 - a) = - a^{2}

How would you solve the following system of linear equations: 2x_1 + (3+a)x_2 + 2x_3 = 2+aZ

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