# Solve for z z/3-1/z=(6z+5)/18

z3-1z=6z+518
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.
3,z,18
Since 3,z,18 contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part 3,1,18 then find LCM for the variable part z1.
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 3 has no factors besides 1 and 3.
3 is a prime number
The number 1 is not a prime number because it only has one positive factor, which is itself.
Not prime
The prime factors for 18 are 2⋅3⋅3.
18 has factors of 2 and 9.
2⋅9
9 has factors of 3 and 3.
2⋅3⋅3
2⋅3⋅3
The LCM of 3,1,18 is 2⋅3⋅3=18.
Multiply 2 by 3.
6⋅3
Multiply 6 by 3.
18
18
The factor for z1 is z itself.
z1=z
z occurs 1 time.
The LCM of z1 is the result of multiplying all prime factors the greatest number of times they occur in either term.
z
The LCM for 3,z,18 is the numeric part 18 multiplied by the variable part.
18z
18z
Multiply each term by 18z and simplify.
Multiply each term in z3-1z=6z+518 by 18z in order to remove all the denominators from the equation.
z3⋅(18z)-1z⋅(18z)=6z+518⋅(18z)
Simplify each term.
Rewrite using the commutative property of multiplication.
18z3z-1z⋅(18z)=6z+518⋅(18z)
Cancel the common factor of 3.
Factor 3 out of 18.
3(6)z3z-1z⋅(18z)=6z+518⋅(18z)
Cancel the common factor.
3⋅6z3z-1z⋅(18z)=6z+518⋅(18z)
Rewrite the expression.
6z⋅z-1z⋅(18z)=6z+518⋅(18z)
6z⋅z-1z⋅(18z)=6z+518⋅(18z)
Multiply z by z by adding the exponents.
Move z.
6(z⋅z)-1z⋅(18z)=6z+518⋅(18z)
Multiply z by z.
6z2-1z⋅(18z)=6z+518⋅(18z)
6z2-1z⋅(18z)=6z+518⋅(18z)
Cancel the common factor of z.
Move the leading negative in -1z into the numerator.
6z2+-1z⋅(18z)=6z+518⋅(18z)
Factor z out of 18z.
6z2+-1z⋅(z⋅18)=6z+518⋅(18z)
Cancel the common factor.
6z2+-1z⋅(z⋅18)=6z+518⋅(18z)
Rewrite the expression.
6z2-1⋅18=6z+518⋅(18z)
6z2-1⋅18=6z+518⋅(18z)
Multiply -1 by 18.
6z2-18=6z+518⋅(18z)
6z2-18=6z+518⋅(18z)
Simplify 6z+518⋅(18z).
Rewrite using the commutative property of multiplication.
6z2-18=186z+518z
Cancel the common factor of 18.
Cancel the common factor.
6z2-18=186z+518z
Rewrite the expression.
6z2-18=(6z+5)z
6z2-18=(6z+5)z
Apply the distributive property.
6z2-18=6z⋅z+5z
Multiply z by z by adding the exponents.
Move z.
6z2-18=6(z⋅z)+5z
Multiply z by z.
6z2-18=6z2+5z
6z2-18=6z2+5z
6z2-18=6z2+5z
6z2-18=6z2+5z
Solve the equation.
Since z is on the right side of the equation, switch the sides so it is on the left side of the equation.
6z2+5z=6z2-18
Move all terms containing z to the left side of the equation.
Subtract 6z2 from both sides of the equation.
6z2+5z-6z2=-18
Combine the opposite terms in 6z2+5z-6z2.
Subtract 6z2 from 6z2.
5z+0=-18
5z=-18
5z=-18
5z=-18
Divide each term by 5 and simplify.
Divide each term in 5z=-18 by 5.
5z5=-185
Cancel the common factor of 5.
Cancel the common factor.
5z5=-185
Divide z by 1.
z=-185
z=-185
Move the negative in front of the fraction.
z=-185
z=-185
z=-185
The result can be shown in multiple forms.
Exact Form:
z=-185
Decimal Form:
z=-3.6
Mixed Number Form:
z=-335
Solve for z z/3-1/z=(6z+5)/18

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