Let's manipulate the expression a little bit and see what we come up with: we have
We can simplify the fraction as
Since both 
and 
are positive, their sum will be positive as well. In other words, we can rewrite the fraction as
So, the sign of this fraction depends on the sign of 
. If its positive, then the whole fraction is positive (product of 4 positive factors). If it's negative, then the whole fraction is negative (product of 3 positive factors and a negative one).
In other words, we arrived to the desired conclusion:
F'(x, y) = 14x +xydy/dx + 14ydy/dx = 0
(xy+14y)|dy/dx = -14x
dy/dx = -14x /(xy+14y)
dy/dx |(x=1,y=1) : -14(1) / (1+14) = -14/15
using
y-y1=/x-x1 = -14/15
y-1/x-1 = -14/15
15y-15 = -14x+14
15y+14x -15-14=0
14x +15y = 29
Plug y= x+5 in 3x+y= 25, we have:
3x+ (x+5)= 25
⇒ (3x+x)+ 5= 25 (combine like terms)
⇒ 4x= 25-5
⇒ 4x= 20
⇒ x= 20/4
⇒x= 5
y= 5+5= 10
Final answer: x=5, y=10~
Let g represent video games. The equation you can use to solve for the amount of video games and CDs Jasmine can buy with her savings is:
43g+72=1128
The 43g part is because it costs 43 dollars for every video game, so you need to multiply 43 by the amount of games she buys to get the total for g games.
The +72 part is because you already have the amount spent on CDs, and needs to be added to the cost of video games, so that the sum would be equal to 1128.
Now, solving:
43g+72=1128 ...subtract 72 from both sides
43g=1056 ...divide both sides by 43 to isolate g
g=24.5581395349
Rounded up, that would be 25 games, but Jasmine's savings isn't enough to buy 25 games that cost $43 each, so you would go down to 24 games.
The answer: Jasmine can buy 24 games with her savings after spending $72 on CDs.
