1y=1p+1q

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

y,p,q

Since y,p,q contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part 1,1,1 then find LCM for the variable part y1,p1,q1.

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 is the result of multiplying all prime factors the greatest number of times they occur in either number.

1

The factor for y1 is y itself.

y1=y

y occurs 1 time.

The factor for p1 is p itself.

p1=p

p occurs 1 time.

The factor for q1 is q itself.

q1=q

q occurs 1 time.

The LCM of y1,p1,q1 is the result of multiplying all prime factors the greatest number of times they occur in either term.

y⋅p⋅q

Multiply yp by q.

ypq

ypq

Multiply each term in 1y=1p+1q by ypq in order to remove all the denominators from the equation.

1y⋅(ypq)=1p⋅(ypq)+1q⋅(ypq)

Cancel the common factor of y.

Factor y out of ypq.

1y⋅(y(pq))=1p⋅(ypq)+1q⋅(ypq)

Cancel the common factor.

1y⋅(y(pq))=1p⋅(ypq)+1q⋅(ypq)

Rewrite the expression.

pq=1p⋅(ypq)+1q⋅(ypq)

pq=1p⋅(ypq)+1q⋅(ypq)

Simplify each term.

Cancel the common factor of p.

Factor p out of ypq.

pq=1p⋅(p(yq))+1q⋅(ypq)

Cancel the common factor.

pq=1p⋅(p(yq))+1q⋅(ypq)

Rewrite the expression.

pq=yq+1q⋅(ypq)

pq=yq+1q⋅(ypq)

Cancel the common factor of q.

Factor q out of ypq.

pq=yq+1q⋅(q(yp))

Cancel the common factor.

pq=yq+1q⋅(q(yp))

Rewrite the expression.

pq=yq+yp

pq=yq+yp

pq=yq+yp

pq=yq+yp

Rewrite the equation as yq+yp=pq.

yq+yp=pq

Factor y out of yq+yp.

Factor y out of yq.

y(q)+yp=pq

Factor y out of yp.

y(q)+y(p)=pq

Factor y out of y(q)+y(p).

y(q+p)=pq

y(q+p)=pq

Divide each term by q+p and simplify.

Divide each term in y(q+p)=pq by q+p.

y(q+p)q+p=pqq+p

Cancel the common factor of q+p.

Cancel the common factor.

y(q+p)q+p=pqq+p

Divide y by 1.

y=pqq+p

y=pqq+p

y=pqq+p

y=pqq+p

Solve for y 1/y=1/p+1/q