Solve for y 1/(7y)-1/(y+7)=1/(2y^2+14y)

Math
17y-1y+7=12y2+14y
Factor 2y out of 2y2+14y.
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Factor 2y out of 2y2.
17y-1y+7=12y(y)+14y
Factor 2y out of 14y.
17y-1y+7=12y(y)+2y(7)
Factor 2y out of 2y(y)+2y(7).
17y-1y+7=12y(y+7)
17y-1y+7=12y(y+7)
Find the LCD of the terms in the equation.
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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
7y,y+7,2y(y+7)
Since 7y,y+7,2y(y+7) 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 7y,y+7,2y(y+7) are:
1. Find the LCM for the numeric part 7,1,2.
2. Find the LCM for the variable part y1,y1.
3. Find the LCM for the compound variable part y+7,y+7.
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.
Since 7 has no factors besides 1 and 7.
7 is a prime number
The number 1 is not a prime number because it only has one positive factor, which is itself.
Not prime
Since 2 has no factors besides 1 and 2.
2 is a prime number
The LCM of 7,1,2 is the result of multiplying all prime factors the greatest number of times they occur in either number.
2⋅7
Multiply 2 by 7.
14
The factor for y1 is y itself.
y1=y
y occurs 1 time.
The LCM of y1,y1 is the result of multiplying all prime factors the greatest number of times they occur in either term.
y
The factor for y+7 is y+7 itself.
(y+7)=y+7
(y+7) occurs 1 time.
The LCM of y+7,y+7 is the result of multiplying all factors the greatest number of times they occur in either term.
y+7
The Least Common Multiple LCM of some numbers is the smallest number that the numbers are factors of.
14y(y+7)
14y(y+7)
Multiply each term by 14y(y+7) and simplify.
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Multiply each term in 17y-1y+7=12y(y+7) by 14y(y+7) in order to remove all the denominators from the equation.
17y⋅(14y(y+7))-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
Simplify 17y⋅(14y(y+7))-1y+7⋅(14y(y+7)).
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Simplify each term.
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Rewrite using the commutative property of multiplication.
1417y(y(y+7))-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
Cancel the common factor of 7.
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Factor 7 out of 14.
7(2)17y(y(y+7))-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
Factor 7 out of 7y.
7(2)17(y)(y(y+7))-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
Cancel the common factor.
7⋅217y(y(y+7))-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
Rewrite the expression.
21y(y(y+7))-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
21y(y(y+7))-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
Combine 2 and 1y.
2y(y(y+7))-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
Cancel the common factor of y.
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Cancel the common factor.
2y(y(y+7))-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
Rewrite the expression.
2(y+7)-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
2(y+7)-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
Apply the distributive property.
2y+2⋅7-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
Multiply 2 by 7.
2y+14-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
Cancel the common factor of y+7.
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Move the leading negative in -1y+7 into the numerator.
2y+14+-1y+7⋅(14y(y+7))=12y(y+7)⋅(14y(y+7))
Factor y+7 out of 14y(y+7).
2y+14+-1y+7⋅((y+7)(14y))=12y(y+7)⋅(14y(y+7))
Cancel the common factor.
2y+14+-1y+7⋅((y+7)(14y))=12y(y+7)⋅(14y(y+7))
Rewrite the expression.
2y+14-1⋅(14y)=12y(y+7)⋅(14y(y+7))
2y+14-1⋅(14y)=12y(y+7)⋅(14y(y+7))
Multiply 14 by -1.
2y+14-14y=12y(y+7)⋅(14y(y+7))
2y+14-14y=12y(y+7)⋅(14y(y+7))
Subtract 14y from 2y.
-12y+14=12y(y+7)⋅(14y(y+7))
-12y+14=12y(y+7)⋅(14y(y+7))
Simplify 12y(y+7)⋅(14y(y+7)).
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Rewrite using the commutative property of multiplication.
-12y+14=1412y(y+7)(y(y+7))
Cancel the common factor of 2.
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Factor 2 out of 14.
-12y+14=2(7)12y(y+7)(y(y+7))
Factor 2 out of 2y(y+7).
-12y+14=2(7)12(y(y+7))(y(y+7))
Cancel the common factor.
-12y+14=2⋅712(y(y+7))(y(y+7))
Rewrite the expression.
-12y+14=71y(y+7)(y(y+7))
-12y+14=71y(y+7)(y(y+7))
Combine 7 and 1y(y+7).
-12y+14=7y(y+7)(y(y+7))
Cancel the common factor of y(y+7).
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Cancel the common factor.
-12y+14=7y(y+7)(y(y+7))
Rewrite the expression.
-12y+14=7
-12y+14=7
-12y+14=7
-12y+14=7
Solve the equation.
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Move all terms not containing y to the right side of the equation.
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Subtract 14 from both sides of the equation.
-12y=7-14
Subtract 14 from 7.
-12y=-7
-12y=-7
Divide each term by -12 and simplify.
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Divide each term in -12y=-7 by -12.
-12y-12=-7-12
Cancel the common factor of -12.
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Cancel the common factor.
-12y-12=-7-12
Divide y by 1.
y=-7-12
y=-7-12
Dividing two negative values results in a positive value.
y=712
y=712
y=712
The result can be shown in multiple forms.
Exact Form:
y=712
Decimal Form:
y=0.583‾
Solve for y 1/(7y)-1/(y+7)=1/(2y^2+14y)

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