OpenStudy (anonymous):

without graphing determine whether it has one solution, no solution or many solutions. 2x+y-5 4x+y=9

OpenStudy (anonymous):

Any guesses or ideas where to start?

OpenStudy (anonymous):

also, should the first expression be "2x+y = 5" instead of "2x+y-5" ? That's not an equation as written... probably just a typo error, but I wanted to be sure.

OpenStudy (anonymous):

yep its 2x+y=5 and i dont know how to start

OpenStudy (anonymous):

subtract the second equationf from first

OpenStudy (anonymous):

You don't have to solve anything here...

OpenStudy (anonymous):

Although you can if that will help

OpenStudy (anonymous):

-2x=-4?

OpenStudy (anonymous):

yes. x = 2

OpenStudy (anonymous):

so one solution??

OpenStudy (anonymous):

A system of 2 line equations like this has exactly one intersection point, UNLESS the lines are actually parallel, in which case they NEVER cross so there is no solution. Lines can't intersect more than once, so "multiple solutions" isn't going to be right.

OpenStudy (anonymous):

Do you know how to know if they are parallel?

OpenStudy (anonymous):

plug in x to find y

OpenStudy (anonymous):

i dont get how thats gonna tell me the answer though :S

OpenStudy (anonymous):

Hint: What is the determinant of the matrix \[\left[\begin{matrix}2 & 1 \\ 4 & 1\end{matrix}\right]\]

OpenStudy (anonymous):

They are parallel if they have the same slope. The slope is the coefficient in front of the x when you put it in y=mx+b form. In this case, one has slope 2 and one has slope 4, so they are NOT parallel, and so they DO cross exactly once

OpenStudy (anonymous):

i never learned derminant or matrix

OpenStudy (anonymous):

If you solve it like @telltoamit or @fredrickV are suggesting, you should get exactly one ordered pair as a solution, and that point is where the two lines cross

OpenStudy (anonymous):

@fredrickV I don't think @kellyking is in matrix algebra yet

OpenStudy (anonymous):

yepp, that means one solution

OpenStudy (anonymous):

Just a question: If the determinant is non-zero, then there is either 1 solution or no solutions correct.

OpenStudy (anonymous):

My point is that with this question, the ONLY thing you have to check is if they are parallel. ALL other lines cross once, i.e. one solution.

OpenStudy (anonymous):

I have to review determinants :) it's been awhile. Don't need them for this question but I should brush up. @fredrickV , you could post your last question as a new question thread so we could get it clarified... just an idea :)

OpenStudy (anonymous):

@kellyking Did you understand it after all that? The conversation went in 2-3 directions :)

OpenStudy (anonymous):

sorry how do you knowif the lines are parallel?

OpenStudy (anonymous):

can u givve me an example with 2 equations

OpenStudy (anonymous):

What I said earlier was wrong. If they are parallel then there are no solutions. If they are the same line then there are infinite solutions.

OpenStudy (anonymous):

use this example... 2x+y=5 -->>> y = -2x+5 4x+y=9 -->>> y = -4x + 9 So the slope of line 1 is -2 and the slope of line 2 is -4, so they will intersect somewhere because they aren't parallel. You would have to solve to find where though.

OpenStudy (anonymous):

If the determinant is 0 then the equations are not linearly independent and there are either no solutions or infinite solutions.

OpenStudy (anonymous):

ohh ok how about if its no solution?

OpenStudy (anonymous):

Yes. The slope.

OpenStudy (anonymous):

Parallel would be like: 4x+y=5 -->>> y = -4x+5 4x+y=9 -->>> y = -4x + 9 Both slopes are -4, so the lines both cross the y axis at different spots, but since the slopes are the same, they never cross each other

OpenStudy (anonymous):

No solution means no intersection, which only happens when they are parallel.

OpenStudy (anonymous):

@fredrickV if they aren't independent, I think it just means you don't really have 2 equations, you have one equation and another "version" of the same. It doesn't tell you what the solution might be though (I think)

OpenStudy (anonymous):

thank you so much!!

OpenStudy (anonymous):

you are quite welcome :)