Addition and Subtraction of Fraction with Variables | Concept Clarification| Solve within Seconds
One must watch this Lecture.This video Helps you to Clear your doubts / Concept in Addition and Subtraction of Fraction with Variables
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On this page, we will learn more about the Addition and subtraction of fractions with variables.
To add or subtract fractions:
You must have a common denominator.
To find the Least Common Denominator (LCD),
take the least common multiple of the individual denominators.
Express each fraction as a new fraction with the common denominator,
by multiplying by one in an appropriate form.
To add fractions with the same denominator:
add the numerators, and keep the denominator the same.
That is, use the rule:
A
C
+
B
C
=
A
+
B
C
Question:
Combine into a single fraction:
2
x
+
3
−
3
x
x
−
1
Solution:
Note that the LCD is
(
x
+
3
)
(
x
−
1
)
.
2
x
+
3
−
3
x
x
−
1
(original expression)
=
2
x
+
3
⋅
x
−
1
x
−
1
−
3
x
x
−
1
⋅
x
+
3
x
+
3
(get a common denominator by multiplying by
1
)
=
2
(
x
−
1
)
−
3
x
(
x
+
3
)
(
x
+
3
)
(
x
−
1
)
(keep the denominator the same; add the numerators)
=
2
x
−
2
−
3
x
2
−
9
x
(
x
+
3
)
(
x
−
1
)
(multiply out the numerator)
=
−
3
x
2
−
7
x
−
2
(
x
+
3
)
(
x
−
1
)
(combine like terms; write numerator in standard form)
To add fractions with different denominators, we must learn how to construct the Lowest Common Multiple of a series of terms.
The Lowest Common Multiple (LCM) of a series of terms
is the smallest product that contains every factor of each term.
For example, consider this series of three terms:
pq pr ps
We will now construct their LCM -- factor by factor.
To begin, it will have the factors of the first term:
LCM = pq
Moving on to the second term, the LCM must have the factors pr. But it already has the factor p -- therefore, we need add only the factor r:
LCM = pqr
Finally, moving on to the last term, the LCM must contain the factors ps. But again it has the factor p, so we need add only the factor s:
LCM = pqrs.
That product is the Lowest Common Multiple of pq, pr, ps. It is the smallest product that contains each of them as factors.
Example 3. Construct the LCM of these three terms: x, x2, x3.
Solution. The LCM must have the factor x.
LCM = x
But it also must have the factors of x2 -- which are x ·x. Therefore, we must add one more factor of x :
LCM = x2
Finally, the LCM must have the factors of x3, which are x· x· x. Therefore,
LCM = x3.
x3 is the smallest product that contains x, x2, and x3 as factors.
We see that when the terms are powers of a variable -- x, x2, x3 -- then their LCM is the highest power.
Problem 2. Construct the LCM of each series of terms.
a) ab, bc, cd. abcd b) pqr, qrs, rst. pqrst
c) a, a2, a3, a4. a4 d) a2b, ab2. a2b2
e) ab, cd. abcd
We will now see what this has to do with adding fraction
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