Solve for m 2-(m-7)/(m-5)=(m+5)/(m^2-5m)-1/m

Math
2-m-7m-5=m+5m2-5m-1m
Factor m out of m2-5m.
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Factor m out of m2.
2-m-7m-5=m+5m⋅m-5m-1m
Factor m out of -5m.
2-m-7m-5=m+5m⋅m+m⋅-5-1m
Factor m out of m⋅m+m⋅-5.
2-m-7m-5=m+5m(m-5)-1m
2-m-7m-5=m+5m(m-5)-1m
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.
1,m-5,m(m-5),m
Since 1,m-5,m(m-5),m 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 1,m-5,m(m-5),m are:
1. Find the LCM for the numeric part 1,1,1,1.
2. Find the LCM for the variable part m1,m1.
3. Find the LCM for the compound variable part m-5,m-5.
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 1,1,1,1 is the result of multiplying all prime factors the greatest number of times they occur in either number.
1
The factor for m1 is m itself.
m1=m
m occurs 1 time.
The LCM of m1,m1 is the result of multiplying all prime factors the greatest number of times they occur in either term.
m
The factor for m-5 is m-5 itself.
(m-5)=m-5
(m-5) occurs 1 time.
The LCM of m-5,m-5 is the result of multiplying all factors the greatest number of times they occur in either term.
m-5
The Least Common Multiple LCM of some numbers is the smallest number that the numbers are factors of.
m(m-5)
m(m-5)
Multiply each term by m(m-5) and simplify.
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Multiply each term in 2-m-7m-5=m+5m(m-5)-1m by m(m-5) in order to remove all the denominators from the equation.
2⋅(m(m-5))-m-7m-5⋅(m(m-5))=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Simplify 2⋅(m(m-5))-m-7m-5⋅(m(m-5)).
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Simplify each term.
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Apply the distributive property.
2⋅(m⋅m+m⋅-5)-m-7m-5⋅(m(m-5))=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Multiply m by m.
2⋅(m2+m⋅-5)-m-7m-5⋅(m(m-5))=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Move -5 to the left of m.
2⋅(m2-5⋅m)-m-7m-5⋅(m(m-5))=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Apply the distributive property.
2m2+2(-5m)-m-7m-5⋅(m(m-5))=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Multiply -5 by 2.
2m2-10m-m-7m-5⋅(m(m-5))=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Cancel the common factor of m-5.
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Move the leading negative in -m-7m-5 into the numerator.
2m2-10m+-(m-7)m-5⋅(m(m-5))=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Factor m-5 out of m(m-5).
2m2-10m+-(m-7)m-5⋅((m-5)m)=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Cancel the common factor.
2m2-10m+-(m-7)m-5⋅((m-5)m)=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Rewrite the expression.
2m2-10m-(m-7)⋅m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
2m2-10m-(m-7)⋅m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Apply the distributive property.
2m2-10m+(-m–7)⋅m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Multiply -1 by -7.
2m2-10m+(-m+7)⋅m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Apply the distributive property.
2m2-10m-m⋅m+7m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Multiply m by m by adding the exponents.
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Move m.
2m2-10m-(m⋅m)+7m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Multiply m by m.
2m2-10m-m2+7m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
2m2-10m-m2+7m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
2m2-10m-m2+7m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Simplify by adding terms.
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Subtract m2 from 2m2.
m2-10m+7m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Add -10m and 7m.
m2-3m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
m2-3m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
m2-3m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Simplify m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5)).
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Simplify each term.
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Cancel the common factor of m(m-5).
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Cancel the common factor.
m2-3m=m+5m(m-5)⋅(m(m-5))-1m⋅(m(m-5))
Rewrite the expression.
m2-3m=m+5-1m⋅(m(m-5))
m2-3m=m+5-1m⋅(m(m-5))
Cancel the common factor of m.
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Move the leading negative in -1m into the numerator.
m2-3m=m+5+-1m⋅(m(m-5))
Cancel the common factor.
m2-3m=m+5+-1m⋅(m(m-5))
Rewrite the expression.
m2-3m=m+5-1⋅(m-5)
m2-3m=m+5-1⋅(m-5)
Apply the distributive property.
m2-3m=m+5-1m-1⋅-5
Rewrite -1m as -m.
m2-3m=m+5-m-1⋅-5
Multiply -1 by -5.
m2-3m=m+5-m+5
m2-3m=m+5-m+5
Simplify by adding terms.
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Combine the opposite terms in m+5-m+5.
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Subtract m from m.
m2-3m=0+5+5
Add 0 and 5.
m2-3m=5+5
m2-3m=5+5
Add 5 and 5.
m2-3m=10
m2-3m=10
m2-3m=10
m2-3m=10
Solve the equation.
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Move 10 to the left side of the equation by subtracting it from both sides.
m2-3m-10=0
Factor m2-3m-10 using the AC method.
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Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is -10 and whose sum is -3.
-5,2
Write the factored form using these integers.
(m-5)(m+2)=0
(m-5)(m+2)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
m-5=0
m+2=0
Set the first factor equal to 0 and solve.
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Set the first factor equal to 0.
m-5=0
Add 5 to both sides of the equation.
m=5
m=5
Set the next factor equal to 0 and solve.
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Set the next factor equal to 0.
m+2=0
Subtract 2 from both sides of the equation.
m=-2
m=-2
The final solution is all the values that make (m-5)(m+2)=0 true.
m=5,-2
m=5,-2
Exclude the solutions that do not make 2-m-7m-5=m+5m2-5m-1m true.
m=-2
Solve for m 2-(m-7)/(m-5)=(m+5)/(m^2-5m)-1/m

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