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What is the strategy to populate a matrix from a linear system of equations?

As a toy example, I have some scary, complicated math, that generates eq1 and eq2. I can solve this, linear, system using Solve. I want to inspect the matrix that is made before the Solve step. In my application I expect this to be a system to be over defined.

eq1 = a x + y;
eq2 = b x - y;
Solve[{eq1 == 7, eq2 == 1}, {x, y}]

In the toy example, it would be this matrix:

( {
   {a, 1},
   {b, -1}
  } ) // MatrixForm

I could imagine parsing the text or something wacky like that.

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CoefficientArrays[{eq1, eq2}, {x, y}][[2]] // Normal

{{a, 1}, {b, -1}}

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