combine like terms

Intro & Vocabulary

      In algebra, you often " combine like terms" -- you sort and collect terms which are like, then write a simplified expression.

      In arithmetic, all terms are alike, because they are all constant. There are no variables in the expression

Examples

      In arithmetic & problem 1.) , all terms are alike, because they are all constant.

      In algebra & problems 2.) and 3.), all terms are not "like", they don't even look alike. So, one combines the ones which are "like," which have exactly the same variable factors.

 

Simplify:

 

1.)    3 + 5 x 2 + 4 (The x here is the multiplication symbol, not a variable.)

2.)    3 + 5x + 4 (Here x is the variable number, x.)

3.)    3 + 5x² + 4x + 4 (Here x is the variable number, x.)

4.)    3 + 5(x + 4) + 4 (Here x is the variable number, x, and one must distribute.)

 

Work & Answer:

1.)   3 + 5 x 2 + 4 is     3 + 10 + 4 is         13 + 4 is         17, the answer.

2.)     3 + 5x + 4 is     3 + 4 + 5x is         7 + 5x, the answer.

3.)     3 + 5x2 + 4x + 4     5x2 + 4x + 4 + 3     5x2 + 4x + 7, the answer.

4.)     3 + 5(x + 4) + 4     3 + 5x + 20 + 4     5x + 20 + 4 + 3     5x + 27, the answer.

 

Exercises

Combine Like Terms

5.) 3x - 6y + 5x - 5y   Answer 8x - 11y

6.) 9x² + 8x - 6x + 4x²   Answer 13x² + 2x

7.) xy² + x²y - 3x²y - xy   Answer xy² - 2x²y - xy   [Hint:

    Terms are like if they have EXACTLY THE SAME VARIABLE FACTORS. The order of the factors does not matter.]

8.) 4(x² - 3x) - 2(x - 2) - (3 - x - x²) Answer 5x² - 13x + 1   [Hint: You must distribute first.]

 

Equation vs. Expression Confusion

      In problems 2, 3, and 4, above, the commutative and associative properties of addition permit one to rearrange the terms so that combining like terms is easy. Each of these problems involves simplifying an expression and combining like terms.

      IN SIMPLIFYING AN EXPRESSION, MOVE TERMS AROUND AS YOU WISH when simplifying an expression and combining like terms.

      IN SIMPLIFYING AN EQUATION,

· WITHOUT MOVING TERMS FROM ONE SIDE TO THE OTHER, MOVE TERMS AROUND AS YOU WISH when simplifying an expression and combining like terms.

· WHEN MOVING TERMS FROM ONE SIDE TO THE OTHER, CHANGE THE SIGN OF THE TERM when simplifying an expression and combining like terms.

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9.)Simplify: 4x + 2 - 3x + 5 - 2(x + 5) 4x + 2 - 3x + 5 - 2x - 10 4x - 3x - 2x + 5 + 2 - 10         - x - 3 , the answer.

10.)Simplify: (2x - 5) + (3x + 4)  2x - 5 +   3x + 4 2x + 3x - 5 + 4 5x - 1, the answer.

11.)Solve: 4x + 2 - 3x + 5 - 2(x + 5) = (2x - 5) + (3x + 4) 4x + 2 - 3x + 5  - 2x - 10 =   2x - 5 +   3x + 4 4x - 3x - 2x + 5 + 2 - 10  =  2x + 3x - 5 + 4         - x             - 3           =       5x       - 1   Thus far each side, each expression, has been simplified seperately. -x - 3 = 5x - 1 +x     = +x      The -x on the left side requires a +x on the right side.    -3   = 6x - 1    +1   =    + 1 The -1 on the left side requires a +1 on the right side.    - 2   = 6x    - 2/6 = 6x/6    - 1/3 = x, the answer

 

More Exercises

12.) Simplify the expression.

4x - 3 + 2x + 3(x + 2)     Answer

13.) Simplify the expression.

(-2 + 5x) - (3x + 1) - 2     Answer

14.) Solve the equation.

4x - 3 + 2x + 3(x + 2) = (-2 + 5x) - (3x + 1) - 2     Answer

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