Notice that one the previous page we used the distributive property
to combine the two terms that had the variable x. The distributive property is the key to combining expressions with like terms.
How can we add and subtract algebraic expressions like 2x + 3x and 4t2 – t2?
To add or subtract algebraic expressions we must have like terms since
2x + 3x = (2 + 3)x = 5x and 4t2 – t2 = (4 – 1)t2 = 3t2.
Why were these statements true? Terms are like terms when they have the exact same variables with the exact same exponents.
Example: 4a,10a, and a are all like terms because in all three expressions the variable is a and the exponent is 1. (Remember that a = a1.)
Example: 27t2, t2, and 96t2 are all like terms because in all four of these expressions, the variable part is t and the exponent on t is 2.
Example: 2a2b, a2b, and 100a2b are all like terms because in all three of these expressions, the variables are a and b, and all the a 's are to the second power and all the b 's are to the first power.
Example: 10xy2 and 17x2 y are not like terms because, although the variables are the same, x does not have the same exponent as x2, and the y2 does not have the same power as y.
In the algebraic expressions we have been looking at, the number that the variable or variables are multiplied by is called the coefficient. For instance, in 10x2y the coefficient part is the 10 and the variable part is the x2y. For t2, the coefficient is 1, since 1t2 = t2.
Once we have like terms, the Distributive Property of Multiplication over Addition allows us to add or subtract the like terms by adding or subtracting the corresponding coefficients. Remember that the Distributive Property of Multiplication over Addition allows us to distribute multiplication over addition or subtraction. If we distribute (4 + 3)x, we get 4x + 3x. (There is a notational convention that says we always put the coefficient in front of the variable). Working backwards, we can see how the distributive property lets us combine the coefficients when adding or subtracting like terms.The variable in the like term is what has been previously distributed.
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Addition Example: |
2x + 3x = (2 + 3)x = 5x |
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Subtraction Example: |
4t2 – t2 = (4 – 1)t2 = 3t2 |
In summary, to add or subtract like terms, we add or subtract the coefficients as indicated, keeping the variable part of the expression the same. This is an application of the Distributive Property of Multiplication over Addition (Subtraction).
Combine like terms for 8n + 7n + n and 9y2 – 5y2 – y.
Each problem that I solved became a rule which served afterwards to solve other problems.
René Descartes (1596–1650)
Discours de la Methode —1637 Discourse on the Method - Wikipedia, the free encyclopedia
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