Solve for t 1/t+1/(t+15)=1/18

1t+1t+15=118
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.
t,t+15,18
Since t,t+15,18 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 t,t+15,18 are:
1. Find the LCM for the numeric part 1,1,18.
2. Find the LCM for the variable part t1.
3. Find the LCM for the compound variable part t+15.
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 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 1,1,18 is 2⋅3⋅3=18.
Multiply 2 by 3.
6⋅3
Multiply 6 by 3.
18
18
The factor for t1 is t itself.
t1=t
t occurs 1 time.
The LCM of t1 is the result of multiplying all prime factors the greatest number of times they occur in either term.
t
The factor for t+15 is t+15 itself.
(t+15)=t+15
(t+15) occurs 1 time.
The LCM of t+15 is the result of multiplying all factors the greatest number of times they occur in either term.
t+15
The Least Common Multiple LCM of some numbers is the smallest number that the numbers are factors of.
18t(t+15)
18t(t+15)
Multiply each term by 18t(t+15) and simplify.
Multiply each term in 1t+1t+15=118 by 18t(t+15) in order to remove all the denominators from the equation.
1t⋅(18t(t+15))+1t+15⋅(18t(t+15))=118⋅(18t(t+15))
Simplify 1t⋅(18t(t+15))+1t+15⋅(18t(t+15)).
Simplify each term.
Rewrite using the commutative property of multiplication.
181t(t(t+15))+1t+15⋅(18t(t+15))=118⋅(18t(t+15))
Combine 18 and 1t.
18t(t(t+15))+1t+15⋅(18t(t+15))=118⋅(18t(t+15))
Cancel the common factor of t.
Cancel the common factor.
18t(t(t+15))+1t+15⋅(18t(t+15))=118⋅(18t(t+15))
Rewrite the expression.
18(t+15)+1t+15⋅(18t(t+15))=118⋅(18t(t+15))
18(t+15)+1t+15⋅(18t(t+15))=118⋅(18t(t+15))
Apply the distributive property.
18t+18⋅15+1t+15⋅(18t(t+15))=118⋅(18t(t+15))
Multiply 18 by 15.
18t+270+1t+15⋅(18t(t+15))=118⋅(18t(t+15))
Rewrite using the commutative property of multiplication.
18t+270+181t+15(t(t+15))=118⋅(18t(t+15))
Combine 18 and 1t+15.
18t+270+18t+15(t(t+15))=118⋅(18t(t+15))
Cancel the common factor of t+15.
Factor t+15 out of t(t+15).
18t+270+18t+15((t+15)t)=118⋅(18t(t+15))
Cancel the common factor.
18t+270+18t+15((t+15)t)=118⋅(18t(t+15))
Rewrite the expression.
18t+270+18t=118⋅(18t(t+15))
18t+270+18t=118⋅(18t(t+15))
18t+270+18t=118⋅(18t(t+15))
36t+270=118⋅(18t(t+15))
36t+270=118⋅(18t(t+15))
Simplify 118⋅(18t(t+15)).
Cancel the common factor of 18.
Factor 18 out of 18t(t+15).
36t+270=118⋅(18(t(t+15)))
Cancel the common factor.
36t+270=118⋅(18(t(t+15)))
Rewrite the expression.
36t+270=t(t+15)
36t+270=t(t+15)
Apply the distributive property.
36t+270=t⋅t+t⋅15
Simplify the expression.
Multiply t by t.
36t+270=t2+t⋅15
Move 15 to the left of t.
36t+270=t2+15t
36t+270=t2+15t
36t+270=t2+15t
36t+270=t2+15t
Solve the equation.
Since t is on the right side of the equation, switch the sides so it is on the left side of the equation.
t2+15t=36t+270
Move all terms containing t to the left side of the equation.
Subtract 36t from both sides of the equation.
t2+15t-36t=270
Subtract 36t from 15t.
t2-21t=270
t2-21t=270
Move 270 to the left side of the equation by subtracting it from both sides.
t2-21t-270=0
Factor t2-21t-270 using the AC method.
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 -270 and whose sum is -21.
-30,9
Write the factored form using these integers.
(t-30)(t+9)=0
(t-30)(t+9)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
t-30=0
t+9=0
Set the first factor equal to 0 and solve.
Set the first factor equal to 0.
t-30=0
Add 30 to both sides of the equation.
t=30
t=30
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
t+9=0
Subtract 9 from both sides of the equation.
t=-9
t=-9
The final solution is all the values that make (t-30)(t+9)=0 true.
t=30,-9
t=30,-9
Solve for t 1/t+1/(t+15)=1/18