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Simplify Expressions

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Simplifying Algebraic Expressions

Corey has a bowl of fruit that consists of 5 apples, 4 oranges, and 3 limes. Katelyn went to the farmer's market and picked up 2 apples, 5 limes, and an orange. How many apples, oranges, and limes do Corey and Katelyn have combined?

Combining like terms is much like grouping together different fruits, like apples and oranges.

Watch This

James Sousa: Combining Like Terms

Guidance

You might have noticed from the previous concept, that sometimes variables and numbers can be repeated within an expression. If the same variable is in an expression more than once, they can be combined by addition or subtraction. This process is called combining like terms.

Example A

Simplify 5x-12-3x+4 .

Solution: Reorganize the expression to group together the x ’s and the numbers. You can either place the like terms next to each together or place parenthesis around the like terms.

& 5x-12-3x+4\\& 5x-3x-12+4 \ or \ (5x-3x)+(-12+4)\\& 2x-8

Notice that the Greatest Common Factor (GCF) for 2x and 8 is 2. Therefore, you can and use the Distributive Property to pull out the GCF to get 2(x-4) .

Example B

Simplify 6a-5b+2a-10b+7

Solution: Here there are two different variables, a and b . Even though they are both variables, they are different variables and cannot be combined. Group together the like terms.

& 6a-5b+2a-10b+7\\& (6a+2a)+(-5b-10b)+7\\& (8a-15b+7)

There is only one number term, called the constant , so we leave it at the end. Also, in general, list the variables in alphabetical order.

Example C

Simplify w^2+9-4w^2+3w^4-7w-11 .

Solution: Here we have one variable, but there are different powers (exponents). Like terms must have the same exponent in order to combine them.

& w^2+9-4w^2+3w^4-7w-11\\& 3w^4+(w^2-4w^2)-7w+(9-11)\\& 3w^4-3w^2-7w-2

When writing an expression with different powers, list the powers from greatest to least, like above.

Intro Problem Revisited Let's rewrite Corey's bowl of fruit as 5a+4o+3l , where a represents apples, o represents oranges, and l represents limes. Then Katelyn's bowl of fruit can be represented as 2a+5l+o . Combining like terms, we have:

(5a+4o+3l)+(2a+5l+o)\\(5a+2a)+(4o+o)+(3l+5l)\\7a+5o+8l

Together they have 7 apples, 5 oranges, and 8 limes.

Guided Practice

Simplify the expressions below.

1. 6s-7t+12t-10s

2. 7y^2-9x^2+y^2-14x+3x^2-4

Answers

1. Combine the s ’s and the t ’s.

& 6s-7t+12t-10s\\& (6s-10s)+(-7t+12t)\\& -4s+5t

2. Group together the like terms.

& 7y^2-9x^2+y^2-14x+3x^2-4\\& (-9x^2+3x^2)+(7y^2+y^2)-14x-4\\& -6x^2+8y^2-14x-4

Notice in #1, we did not write (6s-10s)-(7t+12t) in the second step. This would lead us to an incorrect answer. Whenever grouping together like terms and one is negative (or being subtracted), always change the operator to addition and make the number negative.

In #2, we can also take out the Greatest Common Factor of -2 from each term using the Distributive Property. This would reduce to -2(3x^2-4y^2+7x+2) . In this case, we take out a -2 so that the first term is positive.

Vocabulary

Constant
A number that is added or subtracted within an expression. In the expression 3x^2-8x-15 , -15 is the constant.
Greatest Common Factor (GCF)
The largest number or variable that goes into a set of numbers.

Practice

Simplify the following expressions as much as possible. If the expression cannot be simplified, write “cannot be simplified.”

  1. 5b-15b+8d+7d
  2. 6-11c+5c-18
  3. 3g^2-7g^2+9+12
  4. 8u^2+5u-3u^2-9u+14
  5. 2a-5f
  6. 7p-p^2+9p+q^2-16-5q^2+6
  7. 20x-6-13x+19
  8. 8n-2-5n^2+9n+14

Find the GCF of the following expressions and use the Distributive Property to simplify each one.

  1. 6a-18
  2. 9x^2-15
  3. 14d+7
  4. 3x-24y+21

Challenge We can also use the Distributive Property and GCF to pull out common variables from an expression. Find the GCF and use the Distributive Property to simplify the following expressions.

  1. 2b^2-5b
  2. m^3-6m^2+11m
  3. 4y^4-12y^3-8y^2

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