HSA.REI.C.5 Tutor Intake -- Proving System Equivalence

The equations in the new system look the same as the original equations.

The new system has the same solution set as the original system.

The new system has the same number of equations.

The new equation $E_1 + k \cdot E_2$ is a true statement.

Direction 1 is incorrect; the student needs to restart the proof.

Direction 1 only shows original solutions are in the new system; the
student has not shown that the new system has no extra solutions.

The student needs to verify the proof with a specific numerical example.

The proof requires more than two equations to be complete.

Because specific values would make the arithmetic too complicated.

Because the proof must work for every possible solution, and using
"any solution" proves it for all of them at once.

Because the solution $(3, 1)$ does not satisfy the original system.

Because we do not know what the solution is before solving the system.

$3x + 2y = 9$

$3x = 9$

$x + 2y = 5$

$2x^2 - y^2 = 5$

Replace $E_1$ with $E_1 + 5$ (adding the constant 5 to $E_1$, not a
multiple of another equation).

Replace $E_1$ with $E_1 + 3 E_2$.

Replace $E_1$ with $E_1 \cdot E_2$ (multiplying the two equations).

Replace $E_1$ with $E_1 / x$ (dividing $E_1$ by the variable $x$).