Expert: David Hemmer - 1/2/2011

Question
So I'm a junior in High School and I'm in Advanced Algebra and Trig, & I've been having a little problems with systems.
For example, let me give you a problem.

y=3x+5
2x+4y=6

ok, so I understand that you're supposed to multiply them by a number? & I see that they can be multiplied my 2, 3, whatever number. My problem is that I don't know how I'm supposed to know which number to use. I know that after you multiply you subract?
&..hm I'm just really confused & this is something I should have down by now because finals are next week haha. I just don't understand how I know which number to use. Please help me?
& I need different examples not just that one.

I just really don't understand systems.
I dont understand how you graph them & how you know if a system has no solutions, one solution or infinite solutions.
& for example on our study guide,
this is on there:

2x+y=7
6x+Qy=8

& it says to find a value for Q that would make this have no solution.
Would it be 0? I honestly have no idea.
What would make it have one?

Please help, it would be very highly appreciated.

Answer
Since you hopefully know how to solve problems like 4x+9=14 the goal is to turn your system into an equation like that by eliminating one of the variables. So you have:


(1) y-3x=5
(2) 2x+4y=6.

If you multiply the first equation by 4 you get:

(1') 4y-12x=20.

Now subtract equation (2) from (1') to get:

(4y-12x) - (2x+4y) = 6-20

-14x=-14

x=1.

Once you have x solved for you can plug it to get y. There is nothing special about this. You could have multiplied equation (1) by 2/3 so it's x coefficient is -2. Then add them to cancel off the x. The point is just to get the equations so when you add (or subtract) them one of the variables go's away.

Now for the second half of your question you need to think geometricically what are you doing when you solve these. These are equations of lines so a system of two is just finding the point (s) where two lines intersect. The choices are:

1. No solutions (2 different parallel lines)
2. One solution (not parallel)
3. Infinitely many solutions (the same line!)

So in your sample problem find the slope of the two lines and solve for Q to make them equal.