MHB How to Solve a System of Three Equations with Three Unknowns?

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To solve the system of equations a + b = 3, -b + c = 3, and a + 2c = 10, one approach is to first combine the first two equations to form a new equation: a + c = 6. Next, subtract this new equation from the third equation, a + 2c = 10, to isolate c. Once the value of c is determined, substitute it back into the first equation to find a, and then use either the first or second original equation to calculate b. This method effectively simplifies the problem into a manageable two-variable system. The solution process demonstrates the systematic approach to solving multiple equations with unknowns.
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How do I do:

a + b = 3
-b + c = 3
a + 2c = 10
 
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I would begin by adding the first two equations, and then you have the 2X2 system:

$$a+c=6$$

$$a+2c=10$$

Now, try subtracting the first from the second to get $c$, then use the first along with the value for $c$ to find $a$, and then once you have $a$ and $c$, you can use either the first or second equation from the original system to find $b$. :D

What do you find?
 
Thread 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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