Question

Determine which system below will produce infinitely many solutions.

2x + 5y = 24
2x + 5y = 42
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3x - 2y = 15
6x + 5y = 11
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4x - 3y = 9
-8x + 6y = -18
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5x - 3y = 16
-2x + 3y = -7

Answer 1

@iGreen @sammixboo

can one of yo help me by helping me start off solving the first set if equations? after that I will see about the others and chose my answers n stuff

Answer 2

@iGreen

Answer 3

Well, for it to have infinitely many solutions, they have to be the same line.

Answer 4

how do I know if they are the same line? is it if the equations are the same, but the answer is different? or what

Answer 5

@ganeshie8 ?

Answer 6

Well if we multiply the whole equation by 1 number, it will stay the same.

Like say we have:

3x + 5y = 2

If we multiply this by 6 we get:

18x + 30y = 12

This will be the same line as 3x + 5y = 2.

Answer 7

took me a while but A is parrelel cause they have the same expression but not the same answer

Answer 8

ugh

Answer 9

yes, the two lines given by
\[2x+5y=24\]and\[2x+5y=42\]are parallel and so there are no solutions to that system of equations, as solutions occur only where all of the functions intersect.

Answer 10

Yep, you got it.

Answer 11

Now for B.

Divide 6 / 3 to see if the 2nd equation is the first equation multiplied by a similar number.

Answer 12

I got 6 and 3 from 6x and 3x.

Answer 13

ok i know that

can you help be solve for B?

Answer 14

it is maybe a bit clearer if you consider rearranging them into slope intercept form:

\[2x+5y=24\]solve for \(y\)\[5y=24-2x\]\[y=-\frac{2}{5}x+\frac{24}5\]

\[2x+5y=42\]becomes \[y=-\frac{2}{5}y+\frac{42}{5}\]by similar algebra

Clearly, for any given value of \(x\), the second one gives you a value of \(y\) that is more positive.

Answer 15

B is

3x - 2y = 15
6x + 5y = 11

Answer 16

OS is so slow

Answer 17

i was hoping to use the elimination method, but after I add n stuff idk what to do next

Answer 18

So, if the two lines have different slopes, there will be 1 solution, because they are not parallel and they are not identical, so they will intersect somewhere. That somewhere is the 1 solution.

If the lines have identical slopes, then you need to check the y-intercepts. If the y-intercepts are identical, the two lines are identical and there are infinitely many solutions, as any point on the line works, and a line is made up of an infinite number of points.

If the lines have identical slopes by differing y-intercepts, then they are parallel, and there are no solutions.

Answer 19

see we have

3x - 2y = 15
6x + 5y = 11

add 3x + 6x to get 9x then add -2y + 5y to get 3y then add 15 + 11 to get 26

to we have 9x + 3y = 26

Answer 20

No..that's not how we do it..

Answer 21

or do we multiply -2 by everything in the first equation to get
-6x + 4y = -30
6x + 5y = 11

Answer 22

then we add them together using elimination then get 0 + 9y = -19

Answer 23

That's to get the solution, yes.

Answer 24

@KaylaIsBae Try multiplying -2 to the first equation for Option C.

Answer 25

another way to check is to try multiplying one equation through by a constant to see if you can make the coefficients match those of the other equation. For example:

\[3x+2y = 6\]\[-6x-4y=-12\]If we multiply the first equation by \(-2\), that \(3x\) becomes \(-6x\), just like we have in the other equation. What happens to the rest of it?

\[(-2)3x+(-2)2y=(-2)6\]\[-6x-4y=-12\]
We've converted it to be an exact copy of the second equation. Therefore, these two lines are identical.

Another example:
\[3x+2y=6\]\[-6x-4y=2\]Multiply first equation by \(-2\) giving
\[(-2)3x+(-2)2y=(-2)6\]\[-6x-4y=-12\]That has the same \(x\) and \(y\) coefficients as the second equation, but a different constant. Those lines are parallel.

Remember that the (negative) ratio of the \(x&y\) coefficients forms the slope:

\[Ax+By=C\]\[By=-Ax+C\]\[y=-(\frac{A}{B})x+\frac{C}{A}\]

which when compared with slope intercept form \[y=mx+b\]shows the slope and y-intercept.

Answer 26

i am only supposed to use substitution method or elimination method

Answer 27

Yikes, all those posts interspersed with my stuff...none of that showed up while I was posting!

Answer 28

seethat makes sense but i am still confused. i know what the answer is too, i wanna know how to use these

Answer 29

methods

Answer 30

for B imma try the substitution method tell me if i am wrong

3x - 2y = 15
6x + 5y = 11

taking the first question imma subtract 3x on both sides
3x = 15 + 2y
now dividing 3 on both sides

x = 15/3 + 2y/3
x = 5 + 2y/3

Answer 31

doesn't 2y/3 = 2/3y

Answer 32

uuuuuuuuuuuuuuuuuuuuuuuuuuuuuugh