Question

In the system –12c + 3d = 6 and –3c + 6d = –9, which expression could you use for substitution?

Answer 1

When we solve a system of equations by suvstitution, means that we find an equality for one of the variables and replacing it on the other equation.
Meaning that, our objective, is to turn a equation with two variables into one with a single variable, for example:
\[1) Ax+By=C\]
\[2) A'x+B'x=C'\]
Now, I'll solve equation 1, say for... x, it'll look like this:
\[1)x=\frac{ C-By }{ A }\]
\[2)A'x+B'y=C'\]
Now, I found the exact same value for x, so let's replace it on 2) :
\[2) A'(\frac{ C-By }{ 2 })+B'y=C'\]
\[\frac{ A'C-A'By }{ 2 }+B'y=C'\]
Applying common denomiantor and multiplying both sides by 2:
\[A'C+A'By+2B'y=2C'\]
Reordering a little bit:
\[(A'B+2B')y=2C'-A'C\]
Now, that equation has no meaning at all, it represent almost nothing, but I did it, to show you, and to prove to you, that with this method, we can turn a two variable equation, to a single variable equation, I would've have the same result if I solved it for "y".