Linear Equations with Fractions - Basic Arithmetic

The first step of finding the LCD of a set of fractions is to make sure each of the fractions are simplified. and are already simplified. However, can be reduced to . This makes the problem much easier because we now only have two different denominators to work with. From here, we simply multiply each denominator by increasing integers until we get a common denominator. It is important to always increase the lower of the two denominators. For instance, we have 4 and 3 as denominators in this problem. Since 3 is lower, we will multiply it by 2, getting 6. Now we have 4 and 6. 4 is lower, so we multiply it by 2 to get 8. Now we have 8 and 6. 6 is lower, so we multiply the original denominator of 3 by 3, resulting in denominators of 8 and 9. Following this trend, we get: 12 and 9, then 12 and 12. Therefore, 12 will be the least common denominator.

While simply multiplying all of the denominators will get you a common denominator between the fractions, it does not always give you the LCD.

The correct answer is 36. The best way to approach this problem is to take each denominator and list out its multiples. You can then find the common multiple between each denominator.

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42

Multiples of 12: 12, 24, 36, 48, 60

Multiples of 36: 36, 72, 108

Multiples of 18: 18, 36, 54, 72

From the above multiples, we can see that the least common denominator is 36.

To find the least common denominator, list out the multiples of both denominators until you find the smallest multiple that is shared by both.

4: 4, 8, 12, 16, 20, 24

5: 5, 10, 15, 20, 25

Because 20 is the first shared multiple of 4 and 5, it must be the least common denominator for these two fractions.

To find the least common denominator, list out the multiples of both denominators until you find the smallest multiple that is shared by both.

4: 4, 8, 12, 16, 20, 24, 28, 32

7: 7, 14, 21, 28, 35

Because 28 is the first shared multiple of 4 and 7, it must be the least common denominator for these two fractions.

The least common denominator is the lowest number that has each denominator is a potential factor.

1. Multiply the two lowest denominators:

2. Check to see if the other denominators can factor into that denominator:

So 24 is your least common denominator. (48 is also a common denominator of all of these fractions, but it is not the lowest one)

When finding the least common denominator, the quickest way is to multiply the numbers out.

In this case and are both primes and don't share any factors other than .

We can multiply them to get as the final answer.

Another approach is to list out all the factors of each number and see which factor is in both sets first.

Notice appears in both sets before any other number therefore, this is the least common denominator.

When finding the least common denominator, the quickest way is to multiply the numbers out.

In this case and share a factor other than which is . We can divide those numbers by to get and leftover.

Now, they don't share a common factor so basically multiply them out with the shared factor. Answer is .

Another approach is to list out the factors of both number and find the factor that appears in both sets first.

We can see that appears in both sets before any other number thus, this is our answer.

When finding the least common denominator, the quickest way is to multiply the numbers out. In this case and share a factor other than which is . We can divide those numbers by to get and leftover. Now, they don't share a common factor so basically multiply them out with the shared factor. Answer is .

Another approach is to list out the factors of each number. The factor that appears first in both set is the least common denominator.

We see that appears first in both sets and thus, is the least common denominator.

When finding the least common denominator, the quickest way is to multiply the numbers out. In the case of finding least common denominators among three or more numbers, it's critical there are no common factors between two of the denominators and of course all 3. This will ensure the answer will always be the least common denominator.

Say we just multiplied the numbers out. It's basically or . That number seems big but lets cut this in half and check divides evenly into , , and . Lets check . doesn't divide evenly into so is the answer.

So this goes back to the statement: "In the case of finding least common denominators among three or more numbers, it's critical there are no common factors between two of the denominators and of course all 3." If I factored a , I can reduce the and but not the . That is ok. Now the leftover values are , , and . They only share a factor of . So let's multiply the leftover values and the factored value to get

Start by adding 10 to both sides.

Multiply both side by 9 to get rid of the fraction.

Divide by 5

Since all the answer choices have mixed fractions, you will also need to reduce down to a mixed fraction

Add both sides by 9 to isolate the x on one side.

Multiply both sides by 5.

Divide boths ides b 9.

First, add 10 to both sides so the term with "z" is isolated on one side.

To get rid of the fraction, multiply both sides by 3.

Divide by 2.

Start by adding the terms with together. Find the least common denominator for the two fractions.

Now, multiply both sides by 10.

Then divide both sides by 23.

Start by adding 9 to both sides.

Next, multiply both sides by 3.

Finally, divide both sides by 2.

When we are solving equations, we must always remember that what's on the left side equals the right side. Therefore, any changes that we make to one side of the equation, we must make to the other side so that the equation stays equal and balanced. When solving single-variable equations, we try to isolate the variable on one side so that we can get a number which it's equal to on the other side. Therefore, everything we do to solve this equation must work towards getting just the variable on one side, and a number on the other side. Let's take a look at our equation again:

We want to isolate our term on the left-hand side, so our first step is to get rid of the . To do that, we can perform the inverse operation; we can subtract from both sides:

Now we have our on one side and a number on the other, but we need to see what one is equal to, not . We therefore need to multiply by a number which will make it just . You might remember from fractions that multiplying a fraction by its reciprocal will give you , so let's try multiplying each side by the reciprocal of , which is :

We have one on the left side and a number on the right side, therefore our final answer is