Question

Meaning of: There is only one solution (x,y) and x+y is positive?

Hi as you can see down below I got to the point where I knew there was one solution between A and B, however I didn't understand the the x + y is positive part and to an extent the (x,y) part after one solution. Could anyone help me please, I'd greatly appreciate it!

Answer 1

Answer is `B`.

Explanation:

You probably can't see much from the equations as they stand.
As a start simplify each of the a bit first and see if that helps...
This gives us.
`color(blue)(2/3x+2/3 - 4/5y = 1/3) and color(red)(2/5x+2/5y +1/3 = 1/5)`

not much help! Simplify further..

`color(blue)(2/3x -4/5y = 1/3-2/3 = -1/3)" "larrxx 15`

`color(red)(2/5x +2/5y = 1/5-1/3 = -2/15)" "larrxx 15`

`color(blue)(10x-12y =-5)`
`color(red)(6x+6y = -2)" "larrxx2`
`color(red)(12x+12y = -4`

Now, when we have two equations of the type

`color(blue)(a_1x+b_1y=c_1)` and

`color(red)(a_2x+b_2y=c_2)`

and `a_1/a_2!=b_1/b_2`, then we have only one solution.

If `a_1/a_2=b_1/b_2!=c_1/c_1`, then we have no solution.

and if `a_1/a_2=b_1/b_2=c_1/c_1`, then we have infinite solution.

Here `10/12!=(-12)/12` and hence we have only one solution,

As `12x+12y=-4`

`x+y=-4/12=-1/3`

Hence, answer is `B`.