3/y-2 -1/y=4/y^-2y
3y/y^-2y - y/y^-2y=4/y^-2y
3y-y=4
2y=4
y=2
Put y =2 in the original equation on each side, and see if each side equals the other side.
y=2 should immediately be discarded, since fractions with y-2 in the denominator are not defined.
The steps should be
3y/y(y-2) - (y-2)/y(y-2) = 4/y(y-2)
3y-y+2 = 4
2y = 2
y = 1
To solve the equation 3/y-2 - 1/y = 4/y^(-2y), we need to combine like terms and isolate the variable y.
1. Start by finding a common denominator for the fractions on the left side of the equation. The least common denominator is y^(-2y), so rewrite each fraction with this denominator:
3/y - 2/y = 4/y^(-2y)
2. Combine the fractions using the common denominator:
(3 - 2)/y = 4/y^(-2y)
3. Simplify the numerator:
1/y = 4/y^(-2y)
4. Cross-multiply by multiplying both sides of the equation by y and y^(-2y):
y * 1/y = 4 * y^(-2y)
This simplifies to:
1 = 4/y^(2y)
5. Multiply both sides of the equation by y^(2y) to eliminate the denominator:
1 * y^(2y) = 4 * y^(2y) / y^(2y)
Simplifying further gives us:
y^(2y) = 4
6. At this point, we need to convert the equation into a form that can be solved algebraically. Since we have y raised to a variable power, let's try taking the logarithm of both sides.
Let's take the natural logarithm (ln) of both sides:
ln(y^(2y)) = ln(4)
Applying the logarithmic rule for exponents:
2y * ln(y) = ln(4)
7. Divide both sides of the equation by 2 * ln(y):
2y * ln(y) / (2 * ln(y)) = ln(4) / (2 * ln(y))
Simplifying gives us:
y = ln(4) / (2 * ln(y))
8. At this point, we need to use numerical methods such as graphing, approximation, or iteration to find an estimate for the value of y. It is unlikely that a simple algebraic solution exists.
Therefore, the solution to the equation 3/y-2 - 1/y = 4/y^(-2y) cannot be obtained algebraically, and we will need to use numerical methods to approximate the value of y.