Algebraic Expressions Study Guide: GED Math (page 2)
Practice problems for these concepts can be found at:
What Are Algebraic Expressions?
An algebraic expression is a group of numbers, variables (letters), and operation signs (+, –, ×, ÷). Variables are usually written in italics. For example, the x in the following algebraic expressions is the variable:
Any letter can be used to represent a number in an algebraic expression. The letters x, y, and z are commonly used.
Algebraic expressions translate the relationship between numbers into math symbols.
When multiplying a number and a variable, you just have to write them side by side. You don't need to use a multiplication symbol.
Translate the following math relationship into an algebraic expression: the quotient of 3 divided by a number plus the difference between 3 and 2.
You know that the word quotient indicates division, so write "3 ÷."
The word number represents your variable, so write "3 ÷ x."
The word plus means to add, so write "3 ÷ x +."
The word difference means to subtract, so write: "3 ÷ x + (3 – 2)."
(Remember that parentheses are used to indicate an operation that should be performed first.)
So, the answer is 3 ÷ x + (3 – 2).
When translating a math relationship into an algebraic expression, keep these key words in mind.
Translating Math Relationships into Algebraic Expressions
Remember that the order of the numbers and the exact wording are very important when dividing and subtracting. For example:
- The difference between 3 and 2 means 3 – 2 (which equals 1).
- Three less 2 means 3 – 2 (which equals 1).
- Three less than 2 means 2 – 3 (which equals –1).
- Four divided by 2 means 4 ÷ 2 (which equals 2).
- Two divided by 4 means 2 ÷ 4 (which equals ).
- Two divided into 4 means 4 ÷ 2 (which equals 2).
It's the same when variables are used:
- The difference between 3 and a number means 3 – x.
- Three less than a number means x – 3.
- Four divided by a number means 4 ÷ x.
- A number divided by 4 means x ÷ 4.
Translate the following math relationship into an algebraic expression: one-fourth of a number is decreased by nine.
You know that the word one-fourth is a fraction, so write "."
The word number represents your variable, and when you are talking about fractions, the word of indicates that you should multiply, so write "x."
The words decreased by mean to subtract, so write "x – 9."
The answer is x – 9.
Evaluating Algebraic Expressions
Algebraic expressions have specific values only when the variables have values. Finding the value of an algebraic expression by plugging in the known values of its variables is called evaluating an expression.
- Evaluate 5 ÷ (x + y), when x = 2 and y = 3.
- Find the value of 5 + x, when x = –2. Plug the numbers into the algebraic expression:
Plug the numbers into the algebraic expression: 5 ÷ (2 + 3).
Solve by following the order of operations. Add the numbers in parenthesis: 5 + (2 + 3) = 5 + 5.
Divide to find the final answer: 5 ÷ 5 = 1.
5 + (–2).
Solve: 5 – 2 = 3.
Simplifying Algebraic Expressions
The parts of an algebraic expression are called terms. A term is a number, or a number and the variables associated with it. For example, the following algebraic expression has three terms:
- algebraic expression: 5x2 – 5x + 1
- terms: +5x2, –5x, +1
As you can see, the terms in an algebraic expression are separated by + and – signs. Notice that the sign in front of each term is included as part of that term—the sign is always part of the term. If no sign is given, it is positive (+).
Simplifying an algebraic expression means to combine like terms. Like terms are terms that use the same variable and are raised to the same power. Here are some examples of like terms.
When you combine like terms, you group all the terms that are alike together. This makes it easier to evaluate the expression later on.
- Simplify the following algebraic expression: 5x + 3y + 9x.
- Simplify the following algebraic expression: 2x2 + 3y + 9xy + 3y2.
- Simplify the following algebraic expression: 5x(2x – 1) + 9x.
Write the like terms next to each other. You know that 5x and 9x are like terms because they both have x. So write them next to each other: 5x + 9x +3y.
Combine the like terms. In this case, add them: 5x + 9x + 3y = 14x + 3y.
You can't simplify the 3y because there are no other terms in the expression with the variable y. So you leave it alone. The simplified expression is 14x + 3y.
You cannot write the like terms next to each other, because there are no like terms in this expression. It is already simplified.
Write the like terms next to each other. Begin by distributing the 5x so that you can remove the parentheses. Multiply 5x(2x – 1): 10x2 – 5x.
So now the expression is 10x2 – 5x + 9x.
Now you can see that –5x and 9x are like terms because each contains the x term.
Combine the like terms. Add the like terms: –5x + 9x = 4x.
Now your expression is simplified to 10x2 – 4x.
You can't simplify further because there are no other terms in the expression with the same variable. Therefore, the simplified expression is 10x2 – 4x.
When combining like terms, begin by solving to remove the parentheses. If a negative sign is in front of a set of parentheses, it affects every term inside the parentheses. Here's an example.
–2(x – 3y + 2)
When you multiply each term inside the parentheses by –2, the result is –2x + 6y – 4.
Notice that the negative sign (–) affects each term inside the parentheses.
Practice problems for these concepts can be found at:
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