Question

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

Answer 1

Any guesses or ideas where to start?

Answer 2

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.

Answer 3

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

Answer 4

subtract the second equationf from first

Answer 5

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

Answer 6

Although you can if that will help

Answer 7

-2x=-4?

Answer 8

yes. x = 2

Answer 9

so one solution??

Answer 10

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.

Answer 11

Do you know how to know if they are parallel?

Answer 12

plug in x to find y

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

i never learned derminant or matrix

Answer 17

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

Answer 18

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

Answer 19

yepp, that means one solution

Answer 20

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

Answer 21

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.

Answer 22

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 :)

Answer 23

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

Answer 24

sorry how do you knowif the lines are parallel?

Answer 25

can u givve me an example with 2 equations

Answer 26

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.

Answer 27

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.

Answer 28

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

Answer 29

ohh ok how about if its no solution?

Answer 30

Yes. The slope.

Answer 31

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

Answer 32

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

Answer 33

@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)

Answer 34

thank you so much!!

Answer 35

you are quite welcome :)