Question

(7y-3)/(y-y^2)=1/(y-1)-5/(y-1)

Answer 1

7y-3/y(y-1)=1/(y-1)-5/(y-1)
7y-3=-y+5y
7y-5y+y=3
3y=3
y=3:3
y=1

Answer 2

To solve the given equation, we need to find the value of 'y' that satisfies the equation. Let's proceed with the steps to solve it:

Step 1: Simplify the left-hand side of the equation. Step 2: Combine the fractions on the right-hand side of the equation. Step 3: Set the left-hand side equal to the right-hand side and solve for 'y'. Step 4: Check if the value of 'y' obtained in step 3 satisfies the original equation.

Let's begin:

Step 1: Simplify the left-hand side of the equation. (7y - 3) / (y - y^2)

Step 2: Combine the fractions on the right-hand side of the equation. 1 / (y - 1) - 5 / (y - 1) = (1 - 5) / (y - 1) = -4 / (y - 1)

Step 3: Set the left-hand side equal to the right-hand side and solve for 'y'. (7y - 3) / (y - y^2) = -4 / (y - 1)

To get rid of the denominators, we can cross-multiply:

(7y - 3) * (y - 1) = -4 * (y - y^2)

Expand both sides:

7y^2 - 7y - 3y + 3 = -4y + 4y^2

Combine like terms:

7y^2 - 10y + 3 = 0

Step 4: Solve for 'y' using the quadratic formula or factoring.

Using the quadratic formula: y = (-b ± √(b^2 - 4ac)) / 2a

In our equation: a = 7, b = -10, c = 3

y = [10 ± √((-10)^2 - 4 * 7 * 3)] / (2 * 7)

y = [10 ± √(100 - 84)] / 14

y = [10 ± √16] / 14

y = [10 ± 4] / 14

Now, we have two potential solutions for 'y':

1. y = (10 + 4) / 14 = 14 / 14 = 1
2. y = (10 - 4) / 14 = 6 / 14 = 3 / 7

Step 5: Check if the solutions satisfy the original equation.

For y = 1: (7(1) - 3) / (1 - 1^2) = (7 - 3) / 0 = 4 / 0 (undefined)

For y = 3/7: (7(3/7) - 3) / (3/7 - (3/7)^2) = (21/7 - 3) / (3/7 - 9/49) (21/7 - 21/7) / (3/7 - 9/49) = 0 / (3/7 - 9/49) = 0 / (15/49) = 0

Therefore, the valid solution to the equation is y = 3/7.