Single-Variable Expressions Study Guide (page 2)
Introduction to Single-Variable Expressions
"Obvious" is the most dangerous word in mathematics.
—Eric Temple Bell (1883–1960) Mathematician and Science Fiction Author
In this lesson, you'll review the order of operations and learn how to evaluate algebraic expressions.
The addition sentence 3x + 7y, or even just 3x alone is an algebraic expression. An algebraic expression is one or more terms, at least one of which contains a variable, which may or may not contain an operation (such as addition or multiplication).
We have seen how to add, subtract, multiply, and divide terms, but all of our answers have contained variables. To evaluate an expression, we replace variables with real numbers. At first glance, it might seem easy to evaluate an expression once the variables have been replaced with numbers, but we must remember to follow the order of operations, or we will arrive at the wrong answer.
The order of operations is a list that tells us how to go about evaluating an expression. First, handle any operations that are in parentheses, no matter what those operations are. Next, work with the exponents in the expression. After that, you can do multiplication and division. Finally, perform addition and subtraction. Most people use the acronym PEMDAS to help them remember the order of operations:
P Parentheses P Please
E Exponents E Excuse
M Multiplication M My
D Division D Dear
A Addition A Aunt
S Subtraction S Sally
On the left is a list of the order of operations, and on the right is a phrase (Please Excuse My Dear Aunt Sally) that can help you remember the order of operations. The first letter in each word of the phrase is the same as the first letter in each operation.
To evaluate the numerical expression 6(5) + 4, we multiply 6 and 5 first, since multiplication comes before addition in the order of operations: 6(5) = 30. Then, we add: 30 + 4 = 34. If we were to add 5 and 4 first and then multiply by 6, our answer would be 54, which is incorrect. That's why the order of operations is so important.
2(6 + 4) – 42 =
Begin with the operation in parentheses: 6 + 4 = 10. The expression is now 2(10) – 42.
Next, work with the exponents: 42 = 16, and the expression becomes 2(10) – 16.
Multiplication is next: 2(10) = 20, and we are left with 20 – 16.
Finally, subtract: 20 – 16 = 4. The expression 2(6 + 4) – 42 is equal to 4.
If there is more than one operation inside a set of parentheses, use the order of operations to tell you which operation to perform first. In the expression (5 + 4(3)) – 2, addition and multiplication are both inside parentheses. Because multiplication comes before addition in the order of operations, we begin by multiplying 4 and 3.
Evaluating Single-Variable Expressions
Now that we know how to handle many operations in an expression, we can look at how to evaluate algebraic expressions. The most basic algebraic expressions contain only one variable, and these are called single-variable expressions. When we are given a value for the variable, we can replace that variable with the value (in parentheses).
What is 6x when x = 4?
Replace x in the expression 6x with 4: 6(4) = 24. When x = 4, 6x = 24. But what if x = –10? Then we would replace x with –10: 6(–10) = –60. The value of 6x varies depending on the value of x. This is why x is called a variable!
What is –5p + 9 when p = –1?
Replace p with –1: –5(–1) + 9. Remember the order of operations: Multiply before adding. –5(–1) = 5, 5 + 9 = 14. When p = –1, –5p + 9 is equal to 14.
What is 4(c2 – 7) when c = –3?
Replace c with –3: 4((–3)2 – 7). (–3)2 – 7 is in parentheses, and since exponents come before subtraction, begin there. (–3)2 = 9, 9 – 7 = 2. Finally, 4(2) = 8.
Some expressions may have only one variable, but that variable appears more than once in the expression. For instance, 8x + 2x, or 2(j – 6) – 7j. We have two choices for evaluating these expressions. We can replace every occurrence, or instance, of the variable with its value, or we can simplify the expression first and then replace the variable with its value.
What is 5v – 8v when v = 10?
First, try replacing every v with 10: 5(10) – 8(10).
Multiplication comes before subtraction, so multiply first: 5(10) = 50 and 8(10) = 80.
The expression is now 50 – 80.
Subtract: 50 – 80 = –30.
Now, evaluate the expression again, only this time, simplify it first Remember, to subtract like terms, subtract the coefficient of the second term from the coefficient of the first term, and keep the exponent and base: 5v – 8v = –3v.
Replace v with 10 and multiply: –3(10) = –30.
Either way, we arrive at the same answer. Which way was easier? That is up to you. You might find simplifying first easier, or you might prefer to just substitute the variables with their values right away. As long as you follow the order of operations, both methods will work.
Some expressions cannot be simplified, or can be only partially simplified, often because they do not contain like terms. To evaluate 3x + 3x2 when x= 1, replace both x's with 1 because 3x and 3x2 cannot be combined.
Find practice problems and solutions for these concepts at Single-Variable Expressions Practice Questions.
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