Question

The system of equations graphed below has how many solutions?

http://media.apexlearning.com/Images/200707/07/0bd1ea3b-bcf9-4255-890a-fd029520dcd6.gif

Answer 1

Hi! That link won't work for us. This is probably because you are viewing that image while logged in to your account. You will have to take a screenshot.

Answer 2

You can also type it or draw it.

Answer 3

y=1/2x+1
y=1/2x-4

A. 1
B. 2
C. 0
D. Infinitely many

Answer 4

What course is this, if you don't mind me asking.

Answer 5

Algebra 1

Answer 6

Okay!
So, do you know what they mean by a solution to this "system of equations"?

Answer 7

like how many solutions can it have. I don't really get this yet..

Answer 8

Haha, that's really okay. I asked to find that out. I wouldn't expect you to understand this if you just started learning it. Anything that is so new will take a while to understand. That's okay.

So, if you have a basic equation like
\(6=2x+2\)
It's weird, maybe, but I bet you could find what \(x\) value would make that equation be true. The \(x\) value that makes it true is the "solution."
The solution is \(x=2\) for that. Can you see that?

Answer 9

In a little bit, we'll be able to better understand what a solution for a system of equations is. Right now, we need to make sure we can understand the general idea of what a solution is.

Answer 10

@Sonyalee77 ?

Answer 11

Begin by rewriting the equations so that all of the variables are on one side:
\(\frac{1}{2}x+1 - y =0\)
\(\frac{1}{2}x-4 - y = 0\)

The question is, can we find values for x, y such that all equations in the system are satisfied? Furthermore, can we find an infinite number of solutions?

You will see in this case with two equations the question is equivalent to asking whether or not the lines meet in the graph. If they meet the solution occurs at the point of intersection. In other words the \((x, y)\) pair we were looking for. If they do not meet there is no solution. If they are the same line there is an infinite number of solutions.

Answer 12

sorry computer acting up but so far I get it.

Answer 13

So do you see what the answer is for this particular system?

Answer 14

not yet working on it

Answer 15

Do parallel lines ever meet?

Answer 16

@Alchemista just provided a good synopsis of the entire situation. But we can still build up my example, for fun.

Answer 17

In \(6=2x+2\), the "solution" is the \(x\) value that works!

Now we get a little more complicated.
We'll work with functions like you have probably worked with a lot. Or, just relating one variable (like \(x\)) to another (like \(y\)).
Something like
\(y=2x+2\)
Have you seen lots of these?

Answer 18

Perhaps this might ring a bell:
\(y = mx + b\)

Answer 19

And, parallel lines have the same slope

Answer 20

so is it 0?

Answer 21

yes

Answer 22

YAY!!! Thank you so much @theEric and @Alchemista !!

Answer 23

You're welcome! Feel free to ask if there's anything else about this that you'd like to know!

Answer 24

There is one more much deeper interpretation of this problem I'd like to talk about. You can ignore this if you like.

Let's start by by rewriting the equations as follows
\(\frac{1}{2}x - y = -1\)
\(\frac{1}{2}x - y = 4\)

Notice that the problem is equivalent to asking:
\(x \cdot\left(\begin{array}{cc}
0.5 \\
0.5
\end{array}\right) +
y \cdot\left(\begin{array}{cc}
-1 \\
-1
\end{array}\right) =
\left(\begin{array}{cc}
-1 \\
4
\end{array}\right) \)

So is there a combination (we call this a linear combination) of those two columns which add up to the column at the end.

This begins you on the road to linear algebra.

Answer 25

Thanx \(^_^)/