Subtracting ordinary fractions. How to learn to subtract fractions with different denominators. Summary: general calculation scheme

This article begins the study of operations with algebraic fractions: we will consider in detail such operations as addition and subtraction of algebraic fractions. Let's analyze the scheme for adding and subtracting algebraic fractions with both the same and different denominators. Let's learn how to add an algebraic fraction with a polynomial and how to subtract them. Using specific examples, we will explain each step in finding solutions to problems.

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Addition and subtraction operations with equal denominators

The scheme for adding ordinary fractions is also applicable to algebraic ones. We know that when adding or subtracting common fractions with like denominators, you must add or subtract their numerators, but the denominator remains the same.

For example: 3 7 + 2 7 = 3 + 2 7 = 5 7 and 5 11 - 4 11 = 5 - 4 11 = 1 11.

Accordingly, the rule for adding and subtracting algebraic fractions with like denominators is written in a similar way:

Definition 1

To add or subtract algebraic fractions with like denominators, you need to add or subtract the numerators of the original fractions, respectively, and write the denominator unchanged.

This rule makes it possible to conclude that the result of adding or subtracting algebraic fractions is a new algebraic fraction (in a particular case: a polynomial, monomial or number).

Let us indicate an example of the application of the formulated rule.

Example 1

The algebraic fractions given are: x 2 + 2 · x · y - 5 x 2 · y - 2 and 3 - x · y x 2 · y - 2 . It is necessary to add them.

Solution

The original fractions contain the same denominators. According to the rule, we will perform the addition of the numerators of the given fractions, and leave the denominator unchanged.

Adding the polynomials that are the numerators of the original fractions, we get: x 2 + 2 x y − 5 + 3 − x y = x 2 + (2 x y − x y) − 5 + 3 = x 2 + x y − 2.

Then the required amount will be written as: x 2 + x · y - 2 x 2 · y - 2.

In practice, as in many cases, the solution is given by a chain of equalities, clearly showing all stages of the solution:

x 2 + 2 x y - 5 x 2 y - 2 + 3 - x y x 2 y - 2 = x 2 + 2 x y - 5 + 3 - x y x 2 y - 2 = x 2 + x y - 2 x 2 y - 2

Answer: x 2 + 2 · x · y - 5 x 2 · y - 2 + 3 - x · y x 2 · y - 2 = x 2 + x · y - 2 x 2 · y - 2 .

The result of addition or subtraction can be a reducible fraction, in which case it is optimal to reduce it.

Example 2

It is necessary to subtract the fraction 2 · y x 2 - 4 · y 2 from the algebraic fraction x x 2 - 4 · y 2 .

Solution

The denominators of the original fractions are equal. Let's perform operations with numerators, namely: subtract the numerator of the second from the numerator of the first fraction, and then write the result, leaving the denominator unchanged:

x x 2 - 4 y 2 - 2 y x 2 - 4 y 2 = x - 2 y x 2 - 4 y 2

We see that the resulting fraction is reducible. Let's reduce it by transforming the denominator using the square difference formula:

x - 2 y x 2 - 4 y 2 = x - 2 y (x - 2 y) (x + 2 y) = 1 x + 2 y

Answer: x x 2 - 4 · y 2 - 2 · y x 2 - 4 · y 2 = 1 x + 2 · y.

Using the same principle, three or more algebraic fractions with the same denominators are added or subtracted. Eg:

1 x 5 + 2 x 3 - 1 + 3 x - x 4 x 5 + 2 x 3 - 1 - x 2 x 5 + 2 x 3 - 1 - 2 x 3 x 5 + 2 x 3 - 1 = 1 + 3 x - x 4 - x 2 - 2 x 3 x 5 + 2 x 3 - 1

Addition and subtraction operations with different denominators

Let's look again at the scheme of operations with ordinary fractions: to add or subtract ordinary fractions with different denominators, you need to bring them to a common denominator, and then add the resulting fractions with the same denominators.

For example, 2 5 + 1 3 = 6 15 + 5 15 = 11 15 or 1 2 - 3 7 = 7 14 - 6 14 = 1 14.

Also, by analogy, we formulate the rule for adding and subtracting algebraic fractions with different denominators:

Definition 2

To add or subtract algebraic fractions with different denominators, you must:

• bring the original fractions to a common denominator;
• perform addition or subtraction of resulting fractions with the same denominators.

Obviously, the key here will be the skill of reducing algebraic fractions to a common denominator. Let's take a closer look.

Reducing algebraic fractions to a common denominator

To bring algebraic fractions to a common denominator, it is necessary to carry out an identical transformation of the given fractions, as a result of which the denominators of the original fractions become the same. Here it is optimal to use the following algorithm for reducing algebraic fractions to a common denominator:

• first we determine the common denominator of algebraic fractions;
• then we find additional factors for each of the fractions by dividing the common denominator by the denominators of the original fractions;
• The last action is to multiply the numerators and denominators of the given algebraic fractions by the corresponding additional factors.

The algebraic fractions are given: a + 2 2 · a 3 - 4 · a 2 , a + 3 3 · a 2 - 6 · a and a + 1 4 · a 5 - 16 · a 3 . It is necessary to bring them to a common denominator.

Solution

We act according to the above algorithm. Let's determine the common denominator of the original fractions. For this purpose, we factorize the denominators of the given fractions: 2 a 3 − 4 a 2 = 2 a 2 (a − 2), 3 a 2 − 6 a = 3 a (a − 2) and 4 a 5 − 16 a 3 = 4 a 3 (a − 2) (a + 2). From here we can write the common denominator: 12 a 3 (a − 2) (a + 2).

Now we have to find additional factors. Let us divide, according to the algorithm, the found common denominator into the denominators of the original fractions:

• for the first fraction: 12 · a 3 · (a − 2) · (a + 2) : (2 · a 2 · (a − 2)) = 6 · a · (a + 2) ;
• for the second fraction: 12 · a 3 · (a − 2) · (a + 2) : (3 · a · (a − 2)) = 4 · a 2 · (a + 2);
• for the third fraction: 12 a 3 (a − 2) (a + 2) : (4 a 3 (a − 2) (a + 2)) = 3 .

The next step is to multiply the numerators and denominators of the given fractions by the additional factors found:

a + 2 2 a 3 - 4 a 2 = (a + 2) 6 a (a + 2) (2 a 3 - 4 a 2) 6 a (a + 2) = 6 a (a + 2) 2 12 a 3 (a - 2) (a + 2) a + 3 3 a 2 - 6 a = (a + 3) 4 a 2 ( a + 2) 3 a 2 - 6 a 4 a 2 (a + 2) = 4 a 2 (a + 3) (a + 2) 12 a 3 (a - 2) · (a + 2) a + 1 4 · a 5 - 16 · a 3 = (a + 1) · 3 (4 · a 5 - 16 · a 3) · 3 = 3 · (a + 1) 12 · a 3 (a - 2) (a + 2)

Answer: a + 2 2 · a 3 - 4 · a 2 = 6 · a · (a + 2) 2 12 · a 3 · (a - 2) · (a + 2) ; a + 3 3 · a 2 - 6 · a = 4 · a 2 · (a + 3) · (a + 2) 12 · a 3 · (a - 2) · (a + 2) ; a + 1 4 · a 5 - 16 · a 3 = 3 · (a + 1) 12 · a 3 · (a - 2) · (a + 2) .

So, we have reduced the original fractions to a common denominator. If necessary, you can then convert the resulting result into the form of algebraic fractions by multiplying polynomials and monomials in the numerators and denominators.

Let us also clarify this point: it is optimal to leave the found common denominator in the form of a product in case it is necessary to reduce the final fraction.

We have examined in detail the scheme for reducing initial algebraic fractions to a common denominator; now we can begin to analyze examples of adding and subtracting fractions with different denominators.

Example 4

The algebraic fractions given are: 1 - 2 x x 2 + x and 2 x + 5 x 2 + 3 x + 2. It is necessary to carry out the action of their addition.

Solution

The original fractions have different denominators, so the first step is to bring them to a common denominator. We factor the denominators: x 2 + x = x · (x + 1) , and x 2 + 3 x + 2 = (x + 1) (x + 2) , because roots of a square trinomial x 2 + 3 x + 2 these numbers are: - 1 and - 2. We determine the common denominator: x (x + 1) (x + 2), then the additional factors will be: x+2 And –x for the first and second fractions, respectively.

Thus: 1 - 2 x x 2 + x = 1 - 2 x x (x + 1) = (1 - 2 x) (x + 2) x (x + 1) (x + 2) = x + 2 - 2 x 2 - 4 x x (x + 1) x + 2 = 2 - 2 x 2 - 3 x x (x + 1) (x + 2) and 2 x + 5 x 2 + 3 x + 2 = 2 x + 5 (x + 1) (x + 2) = 2 x + 5 x (x + 1) (x + 2) x = 2 · x 2 + 5 · x x · (x + 1) · (x + 2)

Now let's add the fractions that we have brought to a common denominator:

2 - 2 x 2 - 3 x x (x + 1) (x + 2) + 2 x 2 + 5 x x (x + 1) (x + 2) = = 2 - 2 x 2 - 3 x + 2 x 2 + 5 x x (x + 1) (x + 2) = 2 2 x x (x + 1) (x + 2)

The resulting fraction can be reduced by a common factor x+1:

2 + 2 x x (x + 1) (x + 2) = 2 (x + 1) x (x + 1) (x + 2) = 2 x (x + 2)

And, finally, we write the result obtained in the form of an algebraic fraction, replacing the product in the denominator with a polynomial:

2 x (x + 2) = 2 x 2 + 2 x

Let us write down the solution process briefly in the form of a chain of equalities:

1 - 2 x x 2 + x + 2 x + 5 x 2 + 3 x + 2 = 1 - 2 x x (x + 1) + 2 x + 5 (x + 1) (x + 2 ) = = 1 - 2 x (x + 2) x x + 1 x + 2 + 2 x + 5 x (x + 1) (x + 2) x = 2 - 2 x 2 - 3 x x (x + 1) (x + 2) + 2 x 2 + 5 x x (x + 1) (x + 2) = = 2 - 2 x 2 - 3 x + 2 x 2 + 5 x x (x + 1) (x + 2) = 2 x + 1 x (x + 1) (x + 2) = 2 x (x + 2) = 2 x 2 + 2 x

Answer: 1 - 2 x x 2 + x + 2 x + 5 x 2 + 3 x + 2 = 2 x 2 + 2 x

Pay attention to this detail: before adding or subtracting algebraic fractions, if possible, it is advisable to transform them in order to simplify.

Example 5

It is necessary to subtract fractions: 2 1 1 3 · x - 2 21 and 3 · x - 1 1 7 - 2 · x.

Solution

Let's transform the original algebraic fractions to simplify the further solution. Let's take the numerical coefficients of the variables in the denominator out of brackets:

2 1 1 3 x - 2 21 = 2 4 3 x - 2 21 = 2 4 3 x - 1 14 and 3 x - 1 1 7 - 2 x = 3 x - 1 - 2 x - 1 14

This transformation clearly gave us a benefit: we clearly see the presence of a common factor.

Let's get rid of numerical coefficients in the denominators altogether. To do this, we use the main property of algebraic fractions: we multiply the numerator and denominator of the first fraction by 3 4, and the second by - 1 2, then we get:

2 4 3 x - 1 14 = 3 4 2 3 4 4 3 x - 1 14 = 3 2 x - 1 14 and 3 x - 1 - 2 x - 1 14 = - 1 2 3 x - 1 - 1 2 · - 2 · x - 1 14 = - 3 2 · x + 1 2 x - 1 14 .

Let's perform an action that will allow us to get rid of fractional coefficients: multiply the resulting fractions by 14:

3 2 x - 1 14 = 14 3 2 14 x - 1 14 = 21 14 x - 1 and - 3 2 x + 1 2 x - 1 14 = 14 - 3 2 x + 1 2 x - 1 14 = - 21 · x + 7 14 · x - 1 .

Finally, let’s perform the action required in the problem statement – ​​subtraction:

2 1 1 3 x - 2 21 - 3 x - 1 1 7 - 2 x = 21 14 x - 1 - - 21 x + 7 14 x - 1 = 21 - - 21 x + 7 14 · x - 1 = 21 · x + 14 14 · x - 1

Answer: 2 1 1 3 · x - 2 21 - 3 · x - 1 1 7 - 2 · x = 21 · x + 14 14 · x - 1 .

Adding and subtracting algebraic fractions and polynomials

This action also comes down to adding or subtracting algebraic fractions: it is necessary to represent the original polynomial as a fraction with a denominator 1.

Example 6

It is necessary to add a polynomial x 2 − 3 with the algebraic fraction 3 x x + 2.

Solution

Let's write the polynomial as an algebraic fraction with denominator 1: x 2 - 3 1

Now we can perform addition according to the rule for adding fractions with different denominators:

x 2 - 3 + 3 x x + 2 = x 2 - 3 1 + 3 x x + 2 = x 2 - 3 (x + 2) 1 x + 2 + 3 x x + 2 = = x 3 + 2 · x 2 - 3 · x - 6 x + 2 + 3 · x x + 2 = x 3 + 2 · x 2 - 3 · x - 6 + 3 · x x + 2 = = x 3 + 2 · x 2 - 6 x + 2

Answer: x 2 - 3 + 3 x x + 2 = x 3 + 2 x 2 - 6 x + 2.

If you notice an error in the text, please highlight it and press Ctrl+Enter

Fractions are ordinary numbers and can also be added and subtracted. But because they have a denominator, they require more complex rules than for integers.

Let's consider the simplest case, when there are two fractions with the same denominators. Then:

To add fractions with the same denominators, you need to add their numerators and leave the denominator unchanged.

To subtract fractions with the same denominators, you need to subtract the numerator of the second from the numerator of the first fraction, and again leave the denominator unchanged.

Within each expression, the denominators of the fractions are equal. By definition of adding and subtracting fractions we get:

As you can see, it’s nothing complicated: we just add or subtract the numerators and that’s it.

But even in such simple actions, people manage to make mistakes. What is most often forgotten is that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.

Getting rid of the bad habit of adding denominators is quite simple. Try the same thing when subtracting. As a result, the denominator will be zero, and the fraction will (suddenly!) lose its meaning.

Therefore, remember once and for all: when adding and subtracting, the denominator does not change!

Many people also make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus and where to put a plus.

This problem is also very easy to solve. It is enough to remember that the minus before the sign of a fraction can always be transferred to the numerator - and vice versa. And of course, don’t forget two simple rules:

1. Plus by minus gives minus;
2. Two negatives make an affirmative.

Let's look at all this with specific examples:

Task. Find the meaning of the expression:

In the first case, everything is simple, but in the second, let’s add minuses to the numerators of the fractions:

What to do if the denominators are different

You cannot add fractions with different denominators directly. At least, this method is unknown to me. However, the original fractions can always be rewritten so that the denominators become the same.

There are many ways to convert fractions. Three of them are discussed in the lesson “Reducing fractions to a common denominator”, so we will not dwell on them here. Let's look at some examples:

Task. Find the meaning of the expression:

In the first case, we reduce the fractions to a common denominator using the “criss-cross” method. In the second we will look for the NOC. Note that 6 = 2 · 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are relatively prime. Therefore, LCM(6, 9) = 2 3 3 = 18.

What to do if a fraction has an integer part

I can please you: different denominators in fractions are not the biggest evil. Much more errors occur when the whole part is highlighted in the addend fractions.

Of course, there are own addition and subtraction algorithms for such fractions, but they are quite complex and require a long study. Better use the simple diagram below:

1. Convert all fractions containing an integer part to improper ones. We obtain normal terms (even with different denominators), which are calculated according to the rules discussed above;
2. Actually, calculate the sum or difference of the resulting fractions. As a result, we will practically find the answer;
3. If this is all that was required in the problem, we perform the inverse transformation, i.e. We get rid of an improper fraction by highlighting the whole part.

The rules for moving to improper fractions and highlighting the whole part are described in detail in the lesson “What is a numerical fraction”. If you don’t remember, be sure to repeat it. Examples:

Task. Find the meaning of the expression:

Everything is simple here. The denominators inside each expression are equal, so all that remains is to convert all fractions to improper ones and count. We have:

To simplify the calculations, I have skipped some obvious steps in the last examples.

A small note about the last two examples, where fractions with the integer part highlighted are subtracted. The minus before the second fraction means that the entire fraction is subtracted, and not just its whole part.

Re-read this sentence again, look at the examples - and think about it. This is where beginners make a huge number of mistakes. They love to give such problems on tests. You will also encounter them several times in the tests for this lesson, which will be published shortly.

Summary: general calculation scheme

In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:

1. If one or more fractions have an integer part, convert these fractions to improper ones;
2. Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the writers of the problems did this);
3. Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with like denominators;
4. If possible, shorten the result. If the fraction is incorrect, select the whole part.

Remember that it is better to highlight the whole part at the very end of the task, immediately before writing down the answer.

This lesson will cover adding and subtracting algebraic fractions with different denominators. We already know how to add and subtract common fractions with different denominators. To do this, the fractions must be reduced to a common denominator. It turns out that algebraic fractions follow the same rules. At the same time, we already know how to reduce algebraic fractions to a common denominator. Adding and subtracting fractions with different denominators is one of the most important and difficult topics in the 8th grade course. Moreover, this topic will appear in many topics in the algebra course that you will study in the future. As part of the lesson, we will study the rules for adding and subtracting algebraic fractions with different denominators, and also analyze a number of typical examples.

Let's look at the simplest example for ordinary fractions.

Example 1. Add fractions: .

Solution:

Let's remember the rule for adding fractions. To begin, fractions must be reduced to a common denominator. The common denominator for ordinary fractions is least common multiple(LCM) of the original denominators.

Definition

The smallest natural number that is divisible by both numbers and .

To find the LCM, you need to factor the denominators into prime factors, and then select all the prime factors that are included in the expansion of both denominators.

; . Then the LCM of numbers must include two twos and two threes: .

After finding the common denominator, you need to find an additional factor for each fraction (in fact, divide the common denominator by the denominator of the corresponding fraction).

Each fraction is then multiplied by the resulting additional factor. We get fractions with the same denominators, which we learned to add and subtract in previous lessons.

We get: .

Answer:.

Let us now consider the addition of algebraic fractions with different denominators. First, let's look at fractions whose denominators are numbers.

Example 2. Add fractions: .

Solution:

The solution algorithm is absolutely similar to the previous example. It is easy to find the common denominator of these fractions: and additional factors for each of them.

.

Answer:.

So, let's formulate algorithm for adding and subtracting algebraic fractions with different denominators:

1. Find the lowest common denominator of fractions.

2. Find additional factors for each of the fractions (by dividing the common denominator by the denominator of the given fraction).

3. Multiply the numerators by the corresponding additional factors.

4. Add or subtract fractions using the rules for adding and subtracting fractions with like denominators.

Let us now consider an example with fractions whose denominator contains letter expressions.

Example 3. Add fractions: .

Solution:

Since the letter expressions in both denominators are the same, you should find a common denominator for the numbers. The final common denominator will look like: . Thus, the solution to this example looks like:.

Answer:.

Example 4. Subtract fractions: .

Solution:

If you can’t “cheat” when choosing a common denominator (you can’t factor it or use abbreviated multiplication formulas), then you have to take the product of the denominators of both fractions as the common denominator.

Answer:.

In general, when solving such examples, the most difficult task is to find a common denominator.

Let's look at a more complex example.

Example 5. Simplify: .

Solution:

When finding a common denominator, you must first try to factor the denominators of the original fractions (to simplify the common denominator).

In this particular case:

Then it is easy to determine the common denominator: .

We determine additional factors and solve this example:

Answer:.

Now let's establish the rules for adding and subtracting fractions with different denominators.

Example 6. Simplify: .

Solution:

Answer:.

Example 7. Simplify: .

Solution:

.

Answer:.

Let us now consider an example in which not two, but three fractions are added (after all, the rules of addition and subtraction for a larger number of fractions remain the same).

Example 8. Simplify: .

Note! Before writing your final answer, see if you can shorten the fraction you received.

Subtracting fractions with like denominators, examples:

,

,

Subtracting a proper fraction from one.

If it is necessary to subtract a fraction from a unit that is proper, the unit is converted to the form of an improper fraction, its denominator is equal to the denominator of the subtracted fraction.

An example of subtracting a proper fraction from one:

Denominator of the fraction to be subtracted = 7 , i.e., we represent one as an improper fraction 7/7 and subtract it according to the rule for subtracting fractions with like denominators.

Subtracting a proper fraction from a whole number.

Rules for subtracting fractions - correct from a whole number (natural number):

• We convert given fractions that contain an integer part into improper ones. We obtain normal terms (it doesn’t matter if they have different denominators), which we calculate according to the rules given above;
• Next, we calculate the difference between the fractions that we received. As a result, we will almost find the answer;
• We perform the inverse transformation, that is, we get rid of the improper fraction - we select the whole part in the fraction.

Subtract a proper fraction from a whole number: represent the natural number as a mixed number. Those. We take a unit in a natural number and convert it to the form of an improper fraction, the denominator being the same as that of the subtracted fraction.

Example of subtracting fractions:

In the example, we replaced one with the improper fraction 7/7 and instead of 3 we wrote down a mixed number and subtracted a fraction from the fractional part.

Subtracting fractions with different denominators.

Or, to put it another way, subtracting different fractions.

Rule for subtracting fractions with different denominators. In order to subtract fractions with different denominators, it is necessary, first, to reduce these fractions to the lowest common denominator (LCD), and only after this, perform the subtraction as with fractions with the same denominators.

The common denominator of several fractions is LCM (least common multiple) natural numbers that are the denominators of these fractions.

Attention! If in the final fraction the numerator and denominator have common factors, then the fraction must be reduced. An improper fraction is best represented as a mixed fraction. Leaving the subtraction result without reducing the fraction where possible is an incomplete solution to the example!

Procedure for subtracting fractions with different denominators.

• find the LCM for all denominators;
• put additional factors for all fractions;
• multiply all numerators by an additional factor;
• We write the resulting products into the numerator, signing the common denominator under all fractions;
• subtract the numerators of fractions, signing the common denominator under the difference.

In the same way, addition and subtraction of fractions is carried out if there are letters in the numerator.

Subtracting fractions, examples:

Subtracting mixed fractions.

At subtracting mixed fractions (numbers) separately, the integer part is subtracted from the integer part, and the fractional part is subtracted from the fractional part.

The first option for subtracting mixed fractions.

If the fractional parts the same denominators and numerator of the fractional part of the minuend (we subtract it from it) ≥ numerator of the fractional part of the subtrahend (we subtract it).

For example:

The second option for subtracting mixed fractions.

When fractional parts different denominators. To begin with, we bring the fractional parts to a common denominator, and after that we subtract the whole part from the whole part, and the fractional part from the fractional part.

For example:

The third option for subtracting mixed fractions.

The fractional part of the minuend is less than the fractional part of the subtrahend.

Example:

Because Fractional parts have different denominators, which means, as in the second option, we first bring ordinary fractions to a common denominator.

The numerator of the fractional part of the minuend is less than the numerator of the fractional part of the subtrahend.3 < 14. This means we take a unit from the whole part and reduce this unit to the form of an improper fraction with the same denominator and numerator = 18.

In the numerator on the right side we write the sum of the numerators, then we open the brackets in the numerator on the right side, that is, we multiply everything and give similar ones. We do not open the parentheses in the denominator. It is customary to leave the product in the denominators. We get:

The common denominator of several fractions is the LCM (least common multiple) of the natural numbers that are the denominators of the given fractions.

To the numerators of the given fractions you need to add additional factors equal to the ratio of the LCM and the corresponding denominator.

The numerators of given fractions are multiplied by their additional factors, resulting in numerators of fractions with a single common denominator. Action signs (“+” or “-”) in the recording of fractions reduced to a common denominator are stored before each fraction. For fractions with a common denominator, the action signs are preserved before each reduced numerator.

Only now can you add or subtract the numerators and sign the common denominator under the result.

Attention! If in the resulting fraction the numerator and denominator have common factors, then the fraction must be reduced. It is advisable to convert an improper fraction into a mixed fraction. Leaving the result of an addition or subtraction without canceling the fraction where possible is an incomplete solution to the example!

Adding and subtracting fractions with different denominators. Rule. To add or subtract fractions with different denominators, you must first reduce them to the lowest common denominator, and then perform addition or subtraction as with fractions with the same denominators.

Procedure for adding and subtracting fractions with different denominators

1. find the LCM of all denominators;
2. add additional factors to each fraction;
3. multiply each numerator by an additional factor;
4. take the resulting products as numerators, signing the common denominator under each fraction;
5. add or subtract the numerators of fractions by signing the common denominator under the sum or difference.

Fractions can also be added and subtracted if there are letters in the numerator.