Question

Multiply the equation 2x+2y=8 by 3. Does the equation have the same solution set?

Answer 1

Multiply the equation:

$2x+2y=8 \mapsto 6x+6y=24$

The solution set is the same, because multiplying both sides of an equation by a non-zero number doesn't change the solution set. In fact, if you rewrite the equation as

$2x+2y-8=0$

Multiplying this by 3 (or whatever number, for all it matters) gives

$3(2x+2y-8)=0$

Now, a product is zero if and only if at least one of the factor is zero. So, either $3=0$ or $2x+2y-8=0$

Since the first is clearly impossible, the second one must be true, which is the original equation.