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Chapter 6—Expressions and Equations

6-3 Combining Like Terms

 When you complete the work for this section, you should be able to: Demonstrate how to combine numerical values in an algebraic expression or equation. Demonstrate how to combine literal terms in an algebraic expression or equation. Explain how literal terms can be combined only when they have the same factors raised to the same power

 Definition A term is a part of an expression or equation that appears as a: Numerical value     Example: 2 Literal value (variable)     Example:  x Product of numerical and literal values    Example: 2x

Examples

1. In the expression 2x + 4 - y
2x is a term that is the product of a numerical and literal value
4 is a term that is a numerical value
y is a term that is a literal value

2. In the equation g = 4c – r
g and r are literal-value terms
4c is a product of a numerical and literal term

3. In the equation u = 3 + z/4
u is a  literal value
3 is a numerical value
z/4 is the product of a literal and numerical value (z and 1/4)

 Definition Like terms are terms that have the same literal factors raised to the same power.

Examples

 3x and x The common factors are x. These are like terms y and y/2 The common factors are y. These are like terms x and y Although both are literal terms, they are not they same. These are unlike terms. y and y2 The term y appears in both expressions, but not with the same power. These are unlike terms.

Like terms can also apply to products of literal terms.

 3xy and xy The common factors are x and y. These are like terms hy and hy/2 The common factors are h and y. These are like terms xz and yz The terms share one factor, but not the other. terms, they are not they same. These are unlike terms. zy and zy2 The term y appears in both expressions, but not with the same power. These are unlike terms. 2x2 and 6x2 The literal term is raised to the same power. These are like terms. 4xy2 and xy2 The common factors are x and y2. These are like terms.

Combining Like Terms in Expressions

 An expression such as 2x + 3x means the total of 2 x's and 3 x's. Add them together (as the + sign indicates), and you get 5 x's. In other words, we can say that: 2x + 3x = 5x This is an example of combining like terms. What Unlike terms cannot be combined. Consider the expression, 2x + 3y. This means there are 2 x-things and 3 y-things. The are different things and cannot be combined into a single thing. Of course you can put 2 x-things and 3 y-things into the same container, but you would still have 2x + 3y. These are unlike terms, and they cannot be combined. Rule Only like terms can be combined.

Examples

 Is it possible to combine x + 2x? Is it possible to combine 12 – 5? Is it possible to combine 2x2 + 4x? Is it possible to combine 10y3 + 4y3? Is it possible to combine 6 + x? Yes. x + 2x = 3x Yes. 12 – 5 = 7 No. The x factors have different exponents Yes. 10y3 + 4y3 = 14y3 No. The factors area not the same

What is the purpose for knowing how to combine like terms?  Suppose you are working in a place where your job is to evaluate the same math expression over and over again. The following scenario demonstrates how simplified expressions simplify your work.

Here is the expression you are supposed to evaluate: 4x + 3 + x - 6 - 2x

Suppose this one time you have to evaluate it when x = 5

So substituting 5 for x in the expression:

4(5) + 3 + (5) - 6 - 2(5) = 20 + 3 + 5 - 6 - 10 = 12

I know it isn't terribly difficult, but suppose you have to do it a hundred times with a hundred different values for 5. That is a lot of work. Is there any way to reduce the amount of work? Certainly -- by first simplifying the expression.

So simplify 4x + 3 + x - 6 - 2x by gathering like terms:

4x + 3 + x - 6 - 2x = 3x - 3

The simplified version of 4x + 3 + x - 6 - 2x is 3x - 3

Now, solve the simplified version for x = 5

3(5) - 3 = 15 - 3 = 12

By simplifying expressions, you not only have expressions that are smaller, but require fewer steps when you need to evaluate them.

Examples

Simplify by combining like terms

• 4f - 2g + 6 - 2f simplifies to 2f - 2g + 6

• 3u + 4w + 3u - 6 - 9w simplifies to 6u - 5w - 6

• 2x + y - x/2 + 1 simplifies to 3x/2 +y + 1

Combining Like Terms in Equations

An equation is a set of two expressions separated by an equal sign. You have already seen that you can simplify expressions by combining any like terms; so it figures that the procedure for simplifying equations is a matter of simplifying the terms on each side of the equal sign. That is partly correct. For example:

Simplify this equation:  2x + 4 - x = 6 + 5x - x

Simplifying the expression on the left side of the equal sign:  2x + 4 - x = x + 4

Simplifying the expression on the right side of the equal sign: 6 + 5x - x = 4x  + 6

Setting the two simplified expressions back into the form of an equation:  x + 4 = 4x - 6

The equation is now simpler than before. It's the same equation, but uses fewer terms. It has been simplified ... partly.

Notice there is an x term and a constant term on both sides of the equal sign. The simplification isn't complete until those terms are combined.

 Procedure Combining like terms:  Literal Terms Important:  Only the same literal terms can be combined. When literal terms are on the same side of the equal sign: Combine the like terms by performing the indicated operations When literal terms are on opposite sides of the equal sign: Rewrite the equation so that all the literal terms are on one side of the equal sign (typically the left side). Combine the like terms by performing the indicated operations

Example

 Problem Simplify this equation by combining like terms: x + y = 2 – y Procedure Gather the literal terms on the left side of the equal sign. x + y = 2 – y x + y + y = 2 Perform the indicated operations x + 2y = 2 Solution Combining like terms for x + y = 2 – y, we get x + 2y = 2

Literal terms can be combined only when they have the same power.  For example, 2x2 and x2 can be combined because both terms are raised to the same power power of 2. But 2x2 cannot be combined with x. Why not? Because one of the x terms is raised to the power of 2 and the other is not.

• So 2x2 + x2 can be combined as 3x2
• But 2x2 + x cannot be combined, and must remain expressed as 2x2 + 2x.

Example

 Problem Simplify this equation by combining like terms: 4x2 + 3x + 2 = x2 – x + 3 Procedure Gather the literal terms on the left side of the equal sign. 4x2 + 3x + 2 = x2 – x + 3 4x2 – x2 + 3x + x =  3 – 2 Combine the x2 terms 4x2 – x2 + 3x + x =  3 – 2 3x2 + 3x + x =  3 – 2 Combine the x terms 3x2 + 3x + x =  3 – 2 3x2 + 4x =  3 – 2 Combine the numerical terms 3x2 + 4x =  3 – 2 3x2 + 4x = 1 Solution Simplifying by combining like terms, 4x2 + 3x + 2 = x2 – x + 3 becomes 3x2 + 4x =  1

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