How many solutions does the following system have:
4x + 8y = 16
3x + 6y = 9
Explain how to determine the number of solutions without solving the system. Then apply elimination, and interpret the resulting expression.

- anonymous

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- anonymous

x=2n
y=2-n

- anonymous

you can use matrices?

- anonymous

to find x and y

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## More answers

- anonymous

elimination

- Zarkon

4x + 8y = 16
3x + 6y = 9
givees
x + 2y = 4
x + 2y = 3

- Callisto

4x + 8y = 16 => x+2y=4
3x + 6y = 9 => x+2y =3
they contradict to each other...

- anonymous

3x + 6y = 9=y=3/2-x/2
x=2n+1
y=1-n
y=3-x/2

- anonymous

I'd say put them both into slope intercept form and examine the slopes. If the slopes are different, there is only one solution. If the slopes are the same, examine the vertical intercepts. If the intercepts are the same it has infinitely many solutions, if the intercepts are different it has no solutions.

- anonymous

Without having too much imagination, let the first equation be (1) and the second one (2). Modify the forms in the following way: Let (1)* be (1) divided by four. Let (2)* be (2) divided by three. Combine by subtraction, so (1)*-(2)* gives 0 = 1. This contradiction tells us that the system is inconsistent, and there is no solution.

- anonymous

huh? I have no idea what your saying or what reading

- Callisto

they are // lines, and don't have intercept...

- Callisto

i meant they don't intercept with each other

- anonymous

i know that, i jut don't understand what I'm suppose to do with the equations
everyone else is showing work? but this is the only problem i need help with and dont understand

- anonymous

Do you know what slope-intercept form is?

- anonymous

m= blah blah blah?

- anonymous

Not exactly...
It's y=mx+b

- anonymous

|dw:1332637580537:dw|

- anonymous

oh,

- anonymous

So, your original question asked how to determine the number of solutions WITHOUT solving...

- anonymous

and how do i do that?

- anonymous

Start by isolating "y" in both equations.

- anonymous

4x + 8y = 16
8y+4x=16?

- anonymous

That's a good start. Now, subtract 4x on both sides.

- Callisto

normally, you'll solve the equation .
but when you reduce the equation and find that the slope of the 2 equations is the same. Then you'll have 2 cases, first, then the y-int are not the same, they have no interception point, like this;|dw:1332565127626:dw|If they have the same slope and same y-int, they are of the same line and the equations have infinitely many solutions, like this|dw:1332565207097:dw|
To find the slope and x/y-intercepts you can consider the general form:
ax+by+c=0
slope = -a/b
x-int = -c/a
y-int = -c/b

- anonymous

y+4x=4?

- anonymous

8y+4x=16
-4x -4x

- anonymous

ooo

- anonymous

8y=-4x+16 <---- like this :)

- anonymous

and leave it like that?

- anonymous

Not just yet, we're almost done. Then, to get the y by itself, you can divide each piece by 8.

- anonymous

16/8?

- anonymous

8y=-4x+16
When you divide all 3 terms by 8, you should end up with:
y=-4/8x+16/8
Then you can simplify:
y=-1/2x+2

- anonymous

Now, we're going to write that down somewhere and do the same thing we did to this equation to the other equation. Then we will compare the two.

- anonymous

okay
i think i got it now, thank you figure.

- anonymous

So here's the other equation:
3x + 6y = 9

- anonymous

You got it? Excellent! :)

- anonymous

You're welcome!!

- anonymous

yes, since your the only 1 out of 100 people that make sinces lol

- anonymous

Well, to be fair to the others, they aren't wrong. They're just explaining it at a level that I'm not sure you're ready for just yet. You'll get there though. :)

- anonymous

Thank you, though - of course.

- anonymous

lol i know that, I've never done math higher than pre algebra, I already failed college math 2 times and this is my second and im actually passing with in 2 weeks finally.

- anonymous

lol.

- anonymous

That's excellent! I bet that feels great!

- anonymous

you dont even know. lol

- Zarkon

if you like all the work TheFigure did then you should thank him by pressing the 'Good Answer' button.

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