How do I get the requested result?

Question

I have a statement that says:

Get $a + b + c$, given: $a + b = 15, a + c = 16$

So i can do:

$2a + b + c = 31$, but from here I do not know how to isolate it to get $a + b + c$, I think the exercise is wrong, since there should be 3 equations for 3 variable, right?

Answer 1

There is no unique answer. For example, $\,(a,b,c)=(1,14,15)\,$ and $\,(a,b,c)=(14,1,2)\,$ both satisfy the conditions, but the sums $\,a+b+c\,$ are different in the two cases $\,30 \ne 17\,$.

More generally, $\,(t, 15-t, 16-t)\,$ is a solution for $\,\forall t\,$, but the sum $\,31-t\,$ depends on $\,t\,$.

Answer 2

Yes, $a+b+c$ can take any value.

Note that

$$\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{bmatrix}$$

is nonsingular.

Answer 3

There shouldn't necessarily be $3$ equations, since the question is not to find $a$, $b$ and $c$ but only some linear combination. For instance if you were given $a+b$ and $c$, that's only two equations, but you can still answer.

This being said, it is indeed impossible to find the value of $a+b+c$ given these particular data, as was explained in the other answers.