To solve this equation, we will first simplify both sides of the equation.
Starting with the left side:
6/(y-1) + 2 = y
We can simplify this by first combining the fractions on the left side:
(6 + 2(y-1))/(y-1) = y
Now, simplify further:
(6 + 2y - 2)/(y-1) = y(2y + 4)/(y-1) = y
Moving on to the right side of the equation:
y - (2y + 4 / (1-y))
Simplify the expression by distributing the negative sign:
y - 2y - 4 / (1-y)
Combine like terms:
-y - 4 / (1-y)
Now, we have:
(2y + 4) / (y-1) = y-y - 4 / (1-y) = y
Now, let's solve this equation:
(2y + 4) / (y-1) = y
Cross multiply:
2y + 4 = y(y-1)2y + 4 = y^2 - y
Rearrange to standard form:
y^2 - 3y - 4 = 0
Now let's solve the quadratic equation for y by factoring or using the quadratic formula.
To solve this equation, we will first simplify both sides of the equation.
Starting with the left side:
6/(y-1) + 2 = y
We can simplify this by first combining the fractions on the left side:
(6 + 2(y-1))/(y-1) = y
Now, simplify further:
(6 + 2y - 2)/(y-1) = y
(2y + 4)/(y-1) = y
Moving on to the right side of the equation:
y - (2y + 4 / (1-y))
Simplify the expression by distributing the negative sign:
y - 2y - 4 / (1-y)
Combine like terms:
-y - 4 / (1-y)
Now, we have:
(2y + 4) / (y-1) = y
-y - 4 / (1-y) = y
Now, let's solve this equation:
(2y + 4) / (y-1) = y
Cross multiply:
2y + 4 = y(y-1)
2y + 4 = y^2 - y
Rearrange to standard form:
y^2 - 3y - 4 = 0
Now let's solve the quadratic equation for y by factoring or using the quadratic formula.