How to subtract variables - ISEE Middle Level Quantitative Reasoning

To solve for , the fractions should first be converted to ones that share a common denominator. Given that , the common denominator is 12.

Thus, can be converted to . This gives us:

It is easiest to begin by moving like terms together. Hence:

becomes

(Notice that is its own term.)

Now, consider the coefficients for each term.

For , you have

For , you have

Hence, the expression simplifies to:

This can be moved around to get the correct answer (which means the same thing):

Begin by distributing the two groups. Notice that you must distribute the subtraction through the groups:

becomes

Next, you should move like terms next to each other:

(Notice that is its own term.)

Now, combine terms.

For , you get

For , you get

Therefore, the final form of the expression is:

Begin by distributing. Thus,

becomes

(Don't forget that you have to distribute your subtraction for the second group.)

Combine like terms on the right side of the equation:

Next, move the values to the left side of the equation and all of the other values to the right side:

Combine like terms on the left:

Finally, divide everything by :

This comes out to be:

or

Begin by distributing the multiplied groups:

Next, move all similar factors together:

Now, combine each set of similar factors:

Therefore, our answer is:

This problem is not too difficult. Begin by moving all common terms next to each other:

Next, simplify each group of terms that has the same set of variables:

And do not forget that you are left with as well!

Now, combine all of these:

Begin by moving common factors next to each other. Thus,

becomes

Now, combine each set:

Remember, there still is also.

Therefore, the simplified form of the expression is:

also, since , it follows that

, and by the inequality properties,

making 9 the greater quantity.

When solving this problem we need to remember our order of operations, or PEMDAS.

PEMDAS stands for parentheses, exponents, multiplication/division, and addition/subtraction. When you have a problem with several different operations, you need to solve the problem in this order and you work from left to right for multiplication/division and addition/subtraction.

Parentheses: We are not able to add a variable to a number, so we move to the next step.

Multiplication: We can distribute (or multiply) the .

Addition/Subtraction: Remember, we can't add a variable to a number, so the is left alone.

Subtract the numbers and keep the variable:

Answer:

This problem is as simple as it appears. All that you need to do is group together like terms:

The only like terms are the terms. Therefore, the simple answer is a matter of subtracting 3 from 4:

This problem is just a matter of grouping together like terms. Remember that terms like are treated as though they were their own, different variable:

The only part that might be a little hard is:

If you are confused, think of your number line. This is like "going back" (more negative) from 15. Therefore, you ranswer will be:

This problem really is a trick question. There are no common terms among any of the parts of the expression to be simplified. In each case, you have an independent variable or set of variables: and . Therefore, do not combine any of the elements!

Remember, when there is a subtraction outside of a group, you should add the opposite of each member. That is:

That is a bit confusing, so let's simplify. When you add a negative, you subtract:

Now, group your like variables:

Finally, perform the subtractions and get:

Begin by rewriting the subtracted group as a set of added negative numbers:

Now, simplify that a little by rewriting the additions of negatives as being mere subtractions:

Next, move the like terms next to each other:

Finally, combine like terms:

You need to begin by distributing the minus sign through the whole group . This gives you:

Simplifying the double negative, you get:

Now, you can move the like terms next to each other:

Finally, simplify:

First, start by distributing the subtraction through the terms in parentheses. Note that you will be subtracting negative numbers:

Subtracting a negative is the same as adding a positive:

Now, group the like terms:

All you need to do now is combine like terms:

Begin by distributing the subtraction through the parentheses:

Next, group the like terms:

Now, combine them: