How do you solve the rational equation #5 / (y - 2) = y + 2 #?

Question

Hazel Anglin · Accepted Answer

#color(magenta)(therefore y=3 or y=-3# #5/(y-2)=y+2# #(y+2)(y-2)=5# Identity: #color(red)((a+b)(a-b)=a^2-b^2# #y^2-4=5# #y^2-4-5=0# #y^2-9=0# Identity: #color(red)((a+b)(a-b)=a^2-b^2# #(y+3)(y-3)=0# #therefore y-3=0 or y+3=0# #color(magenta)(therefore y=3 or y=-3#

Savannah Anderson · Answer

#y=+-3# #"multiply both sides by "(y-2)# #cancel((y-2))xx5/cancel((y-2))=(y-2)(y+2)# #rArr(y-2)(y+2)=5larrcolor(blue)"expand factors"# #rArry^2-4=5larrcolor(blue)"add 4 to both sides"# #rArry^2=9# #color(blue)"take square root of both sides"# #rArry=+-3larrcolor(blue)"note plus or minus"# #color(blue)"As a check"# #y=3to5/(3-2)=3+2to5=5" True"# #y=-3to5/(-3-2)=-3+2to-1=-1" True"# #rArry=+-3" are the solutions"#

Isaiah Anthony · Answer

undefined To solve the rational equation 5 / (y - 2) = y + 2, you can start by multiplying both sides of the equation by (y - 2) to eliminate the denominator. This gives you 5 = (y + 2)(y - 2). Expanding the right side of the equation, you get 5 = y^2 - 4. Rearranging the equation, you have y^2 - 4 - 5 = 0. Simplifying further, you get y^2 - 9 = 0. Factoring the quadratic equation, you have (y - 3)(y + 3) = 0. Setting each factor equal to zero, you find y = 3 or y = -3. Therefore, the solutions to the rational equation are y = 3 and y = -3.