-2x-4y=1 and 12y=-6x-3 what is the solution

Question

whpalmer4 · Answer

Can you rearrange one of the equations so that they have the variables on the same side of the equals sign, and in the same order?

anonymous · Answer

There are actually two ways to solve it: 
do what whpalmer4 said:
Rearrange both equations so that they both equal y 
y = ?
y = ?
And then apply both y's together
y = y
? = ?
and solve x

OR

Rearrange one of them and substitute it into the other equation

whpalmer4 · Answer

There are many ways to solve this, not just two.  

$$-2x-4y=1$$$$12y=-6x-3$$We can rearrange the second equation:
$$12y=-6x-3$$$$12y+6x=-3$$$$6x+12y=-3$$

Now write the two equations together:
$$-2x-4y=1$$$$6x+12y=-3$$

Next, we multiply one of the equations by a number that will make it have the same coefficient in one of the columns as the other equation has:

$$3(-2x-4y=1)$$$$6x+12y=-3$$

$$-6x-12y=3$$$$6x+12y=-3$$
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Now we add the equations together, and something interesting happens...

$$-6x+6x -12y+12y = 3-3$$$$0=0$$That means that the two lines are in fact identical lines, so each point on the line (and there are of course an infinite number of them) is a solution to the system of equations!

If we had gotten $0=1$ or some other nonsensical statement, that would mean the lines are parallel and never intersect—no solutions.

The 3rd case, and probably the most likely case for most problems, is that we end up with an equation in terms of just one variable, which we can solve.  Then we plug that value into the original equation and find the value of the other variable.  This case represents the lines intersecting (which they will do only once if they are straight lines).