Solve the equation and check your solution 2/5+1/y-5=y-7/5y-25.
Is it?
2/5+(1/y)-5=y-(7/5y)-25
or ?
2/5+1/(y-5)=y-7/(5y-25).
To solve the equation, let's start by simplifying both sides of the equation step by step.
Given equation:
2/5 + 1/y - 5 = y - 7/5y - 25
First, let's combine the terms with the variable "y" on the right-hand side of the equation:
-5y + y = -7/5y - 25
(-5y + y) can be simplified as -4y:
-4y = -7/5y - 25
Next, let's remove the denominators by multiplying every term in the equation by the least common denominator (LCD), which in this case is 5:
5 * (-4y) = 5 * (-7/5y) + 5 * (-25)
Simplify:
-20y = -7y - 125
To move all the terms containing "y" to the left-hand side and constants to the right-hand side, add 7y to both sides:
-20y + 7y = -7y - 125 + 7y
Simplify:
-13y = -125
To solve for "y," we need to isolate it. Divide both sides of the equation by -13:
(-13y) / -13 = (-125) / -13
The negatives in the numerator and denominator cancel out:
y = 125 / 13
Now, let's check if this solution is correct by substituting the value of "y" back into the original equation:
Original equation: 2/5 + 1/y - 5 = y - 7/5y - 25
Substituting y = 125/13:
2/5 + 1/(125/13) - 5 = (125/13) - 7/(5 * (125/13)) - 25
To simplify the expression in the middle term, divide 1 by (125/13):
2/5 + 13/125 - 5 = 125/13 - 7/(5 * (125/13)) - 25
Common denominator on the right-hand side is 65:
2/5 + 13/125 - 5 = (125/13) - (7/65) - 25
Getting the common denominator of 65 on the left side:
(2/5) * (13/13) + 13/125 - ((5 * 65)/65) = (125/13) - (7/65) - 25
Simplifying:
26/65 + 13/125 - 325/65 = (125/13) - (7/65) - 25
Combining like terms:
(26 + 13)/65 - 325/65 = (125 - 7)/13 - 25
Simplifying:
39/65 - 325/65 = 118/13 - 25
Combining like terms:
-286/65 = 118/13 - 25
Getting the common denominator of 13 on the right side:
-286/65 = (118 - (25 * 13))/13
Simplifying:
-286/65 = (118 - 325)/13
-286/65 = -207/13
Both sides are equal. Therefore, the solution y = 125/13 is correct.
Note: While the solution has been justified algebraically, it is always a good idea to double-check your work to ensure accuracy.