Solve basic system of equations with two unknowns. Why solution doesn't exists?

Question

The following equation has no solution, but I'm not sure why exactly: $7 = 3x_1 + 5x_2 \\ 13 = (1 - x_1)3 + (1 - x_2)5 + 21$

Code for Wolfram Alpha

7  = 3*x_1 + 5*x_2
13 = (1 - x_1) * 3 + (1 - x_2) * 5 + 21

It seems there are two unknowns and two equations, so this seems like something which should be solvable?

I appreciate that it isn't, but I'm not sure how to explain to myself why it isn't solvable. I just remember that if there were $n$ equations and $\leq n$ unknowns, we were able to solve the system .

Answer 1

If you rewrite your system as a linear equation, you end up with a singular matrix of rank 1. The column space (image space) of a rank 1 matrix does not span $\mathbb{R}^2$ (it is one-dimensional), so almost every right hand side (or left hand side in your case) will result in a system without a solution.

Put differently: The two equations on your right hand side are not linearly independent (they are the same up to linear transformations), but your left hand side does not reflect that.

Answer 2

It seems the equations are nonsensical, since expanding out the second one yields $ \sout{5x_2 = - 3x_1}$ which gives $7 = 0$ when substituted back into the first equation. The equations are inconsistent with/incongruent to each other.

Sorry, strike that. Missed a minus sign. But 16 still does not equal 7 so the comment holds.