# Solve for y 12/y=6/(2y-6)+1 12y=62y-6+1
Factor each term.
Factor 2 out of 2y-6.
Factor 2 out of 2y.
12y=62(y)-6+1
Factor 2 out of -6.
12y=62y+2⋅-3+1
Factor 2 out of 2y+2⋅-3.
12y=62(y-3)+1
12y=62(y-3)+1
Reduce the expression 62(y-3) by cancelling the common factors.
Factor 2 out of 6.
12y=2⋅32(y-3)+1
Cancel the common factor.
12y=2⋅32(y-3)+1
Rewrite the expression.
12y=3y-3+1
12y=3y-3+1
12y=3y-3+1
Find the LCD of the terms in the equation.
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
y,y-3,1
Since y,y-3,1 contain both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.
Steps to find the LCM for y,y-3,1 are:
1. Find the LCM for the numeric part 1,1,1.
2. Find the LCM for the variable part y1.
3. Find the LCM for the compound variable part y-3.
4. Multiply each LCM together.
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number 1 is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of 1,1,1 is the result of multiplying all prime factors the greatest number of times they occur in either number.
1
The factor for y1 is y itself.
y1=y
y occurs 1 time.
The LCM of y1 is the result of multiplying all prime factors the greatest number of times they occur in either term.
y
The factor for y-3 is y-3 itself.
(y-3)=y-3
(y-3) occurs 1 time.
The LCM of y-3 is the result of multiplying all factors the greatest number of times they occur in either term.
y-3
The Least Common Multiple LCM of some numbers is the smallest number that the numbers are factors of.
y(y-3)
y(y-3)
Multiply each term by y(y-3) and simplify.
Multiply each term in 12y=3y-3+1 by y(y-3) in order to remove all the denominators from the equation.
12y⋅(y(y-3))=3y-3⋅(y(y-3))+1⋅(y(y-3))
Simplify 12y⋅(y(y-3)).
Cancel the common factor of y.
Cancel the common factor.
12y⋅(y(y-3))=3y-3⋅(y(y-3))+1⋅(y(y-3))
Rewrite the expression.
12⋅(y-3)=3y-3⋅(y(y-3))+1⋅(y(y-3))
12⋅(y-3)=3y-3⋅(y(y-3))+1⋅(y(y-3))
Apply the distributive property.
12y+12⋅-3=3y-3⋅(y(y-3))+1⋅(y(y-3))
Multiply 12 by -3.
12y-36=3y-3⋅(y(y-3))+1⋅(y(y-3))
12y-36=3y-3⋅(y(y-3))+1⋅(y(y-3))
Simplify 3y-3⋅(y(y-3))+1⋅(y(y-3)).
Simplify each term.
Cancel the common factor of y-3.
Factor y-3 out of y(y-3).
12y-36=3y-3⋅((y-3)y)+1⋅(y(y-3))
Cancel the common factor.
12y-36=3y-3⋅((y-3)y)+1⋅(y(y-3))
Rewrite the expression.
12y-36=3⋅y+1⋅(y(y-3))
12y-36=3⋅y+1⋅(y(y-3))
Multiply y(y-3) by 1.
12y-36=3y+y(y-3)
Apply the distributive property.
12y-36=3y+y⋅y+y⋅-3
Multiply y by y.
12y-36=3y+y2+y⋅-3
Move -3 to the left of y.
12y-36=3y+y2-3y
12y-36=3y+y2-3y
Combine the opposite terms in 3y+y2-3y.
Subtract 3y from 3y.
12y-36=y2+0
12y-36=y2
12y-36=y2
12y-36=y2
12y-36=y2
Solve the equation.
Subtract y2 from both sides of the equation.
12y-36-y2=0
Factor the left side of the equation.
Factor -1 out of 12y-36-y2.
Reorder the expression.
Move -36.
12y-y2-36=0
Reorder 12y and -y2.
-y2+12y-36=0
-y2+12y-36=0
Factor -1 out of -y2.
-(y2)+12y-36=0
Factor -1 out of 12y.
-(y2)-(-12y)-36=0
Rewrite -36 as -1(36).
-(y2)-(-12y)-1⋅36=0
Factor -1 out of -(y2)-(-12y).
-(y2-12y)-1⋅36=0
Factor -1 out of -(y2-12y)-1(36).
-(y2-12y+36)=0
-(y2-12y+36)=0
Factor using the perfect square rule.
Rewrite 36 as 62.
-(y2-12y+62)=0
Check the middle term by multiplying 2ab and compare this result with the middle term in the original expression.
2ab=2⋅y⋅-6
Simplify.
2ab=-12y
Factor using the perfect square trinomial rule a2-2ab+b2=(a-b)2, where a=y and b=-6.
-(y-6)2=0
-(y-6)2=0
-(y-6)2=0
Multiply each term in -(y-6)2=0 by -1
Multiply each term in -(y-6)2=0 by -1.
(-(y-6)2)⋅-1=0⋅-1
Multiply (-(y-6)2)⋅-1.
Multiply -1 by -1.
1(y-6)2=0⋅-1
Multiply (y-6)2 by 1.
(y-6)2=0⋅-1
(y-6)2=0⋅-1
Multiply 0 by -1.
(y-6)2=0
(y-6)2=0
Set the y-6 equal to 0.
y-6=0
Add 6 to both sides of the equation.
y=6
y=6
Solve for y 12/y=6/(2y-6)+1

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