Solve for y 1/(y+10)+1/y=1/21

Math
1y+10+1y=121
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.
y+10,y,21
Since y+10,y,21 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+10,y,21 are:
1. Find the LCM for the numeric part 1,1,21.
2. Find the LCM for the variable part y1.
3. Find the LCM for the compound variable part y+10.
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
21 has factors of 3 and 7.
3⋅7
Multiply 3 by 7.
21
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+10 is y+10 itself.
(y+10)=y+10
(y+10) occurs 1 time.
The LCM of y+10 is the result of multiplying all factors the greatest number of times they occur in either term.
y+10
The Least Common Multiple LCM of some numbers is the smallest number that the numbers are factors of.
21y(y+10)
21y(y+10)
Multiply each term by 21y(y+10) and simplify.
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Multiply each term in 1y+10+1y=121 by 21y(y+10) in order to remove all the denominators from the equation.
1y+10⋅(21y(y+10))+1y⋅(21y(y+10))=121⋅(21y(y+10))
Simplify 1y+10⋅(21y(y+10))+1y⋅(21y(y+10)).
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Simplify each term.
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Rewrite using the commutative property of multiplication.
211y+10(y(y+10))+1y⋅(21y(y+10))=121⋅(21y(y+10))
Combine 21 and 1y+10.
21y+10(y(y+10))+1y⋅(21y(y+10))=121⋅(21y(y+10))
Cancel the common factor of y+10.
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Factor y+10 out of y(y+10).
21y+10((y+10)y)+1y⋅(21y(y+10))=121⋅(21y(y+10))
Cancel the common factor.
21y+10((y+10)y)+1y⋅(21y(y+10))=121⋅(21y(y+10))
Rewrite the expression.
21y+1y⋅(21y(y+10))=121⋅(21y(y+10))
21y+1y⋅(21y(y+10))=121⋅(21y(y+10))
Rewrite using the commutative property of multiplication.
21y+211y(y(y+10))=121⋅(21y(y+10))
Combine 21 and 1y.
21y+21y(y(y+10))=121⋅(21y(y+10))
Cancel the common factor of y.
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Cancel the common factor.
21y+21y(y(y+10))=121⋅(21y(y+10))
Rewrite the expression.
21y+21(y+10)=121⋅(21y(y+10))
21y+21(y+10)=121⋅(21y(y+10))
Apply the distributive property.
21y+21y+21⋅10=121⋅(21y(y+10))
Multiply 21 by 10.
21y+21y+210=121⋅(21y(y+10))
21y+21y+210=121⋅(21y(y+10))
Add 21y and 21y.
42y+210=121⋅(21y(y+10))
42y+210=121⋅(21y(y+10))
Simplify 121⋅(21y(y+10)).
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Cancel the common factor of 21.
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Factor 21 out of 21y(y+10).
42y+210=121⋅(21(y(y+10)))
Cancel the common factor.
42y+210=121⋅(21(y(y+10)))
Rewrite the expression.
42y+210=y(y+10)
42y+210=y(y+10)
Apply the distributive property.
42y+210=y⋅y+y⋅10
Simplify the expression.
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Multiply y by y.
42y+210=y2+y⋅10
Move 10 to the left of y.
42y+210=y2+10y
42y+210=y2+10y
42y+210=y2+10y
42y+210=y2+10y
Solve the equation.
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Since y is on the right side of the equation, switch the sides so it is on the left side of the equation.
y2+10y=42y+210
Move all terms containing y to the left side of the equation.
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Subtract 42y from both sides of the equation.
y2+10y-42y=210
Subtract 42y from 10y.
y2-32y=210
y2-32y=210
Move 210 to the left side of the equation by subtracting it from both sides.
y2-32y-210=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=1, b=-32, and c=-210 into the quadratic formula and solve for y.
32±(-32)2-4⋅(1⋅-210)2⋅1
Simplify.
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Simplify the numerator.
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Raise -32 to the power of 2.
y=32±1024-4⋅(1⋅-210)2⋅1
Multiply -210 by 1.
y=32±1024-4⋅-2102⋅1
Multiply -4 by -210.
y=32±1024+8402⋅1
Add 1024 and 840.
y=32±18642⋅1
Rewrite 1864 as 22⋅466.
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Factor 4 out of 1864.
y=32±4(466)2⋅1
Rewrite 4 as 22.
y=32±22⋅4662⋅1
y=32±22⋅4662⋅1
Pull terms out from under the radical.
y=32±24662⋅1
y=32±24662⋅1
Multiply 2 by 1.
y=32±24662
Simplify 32±24662.
y=16±466
y=16±466
The final answer is the combination of both solutions.
y=16+466,16-466
y=16+466,16-466
The result can be shown in multiple forms.
Exact Form:
y=16+466,16-466
Decimal Form:
y=37.58703314…,-5.58703314…
Solve for y 1/(y+10)+1/y=1/21

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