Question

Real numbers x and y satisfy x + xy^2 = 250y, x - xy^2 = -240y. Enter all possible values of x, separated by commas.

Answer 1

Answer:

-35,35,0

Step-by-step explanation:

Let's organise our information :

• x+xy²=250y
• x-xy²= -240y

Now let's add them together :

• x+xy²+x-xy²=250y-240y
• 2x=10y
• x=5y ⇒ y= x/5

Let's replace y by x/5 in the first equation :

• x+x*(x/5)²=250*(x/5)
• x+ (x∧3)/25 = 50x
• 49x-(x∧3)/25=0
• x(49-(x²/25))=0
• x=0 or 49-(x²/25)=0
• x=0 or 49= x²/25
• x=0 or x²=1225
• x=0 or  x= $\sqrt{1225}$ or x= -$\sqrt{1225\\}$
• x=0 or x=35 or x= -35

so x=[0,35,(-35)]

Answer 2

Answer: x = {0, 35, -35}

Step-by-step explanation:

Add the two equations together to eliminate the xy² term.

     x + xy² = 250y

+    x - xy² = -240y

    2x        =  10y

  ÷2             ÷2  

      x        =  5y

 

Substitute x = 5y into either of the equations to solve for "y".

5y + 5y(y²) = 250y

5y + 5y³ = 250y

5y³ + 5y - 250y = 0

5y³ - 245y = 0

5y(y² - 49) = 0

5y = 0     y² - 49 = 0

y = 0       y = ±7

Substitute y = 0, y = 7, and y = -7 into either of the equations to solve for "x".

x + x(0)² = 250(0)          x + x(7)² = 250(7)        x + x(-7)² = 250(-7)

x +    0    =   0                x + 49x = 1750            x + 49x = -1750

        x    =    0                      50x = 1750               50x = -1750

                                                x = 35                       x = -35