Overview
A system asks for values that satisfy both equations at once. Substitution replaces one variable with an equivalent expression from the other equation. Elimination adds suitable multiples of equations so one variable cancels. Neither method changes the common solutions when every operation is applied to a whole equation.
Consider this example: If x + y = 9 and x − y = 3, what is x? Add the equations to eliminate y: 2x = 12. Thus x = 6; the corresponding y = 3 satisfies both equations. The value 3 is y, not the requested x.
The method also handles a different case: If y = 2x − 1 and x + y = 11, what is y? Substitute 2x − 1 for y: 3x − 1 = 11, so x = 4. Then y = 2(4) − 1 = 7. Choosing 4 stops at the other coordinate.
For a further application, consider: If x + 2y = 7 and 3x + ay = 9, with a ≠ 6, what is y? Subtract 3 times the first equation from the second: (a − 6)y = 9 − 21 = −12. Since a ≠ 6, y = −12/(a − 6). At a = 6 the system would demand 0 = −12, so the excluded parameter cannot be restored.
A pair that satisfies just one equation is not a system solution. Multiplying only selected terms during elimination creates a different equation. Choose the method with the least rewriting: an isolated variable favors substitution, while opposite coefficients favor elimination. Substitute the final pair into both original equations.