Math Academy

Identifying Terms in Linear Expressions

Listing the Terms of a Linear Expression

Identifying Particular Terms of an Expression

Prerequisites

Introduction

Let's take a look at the following algebraic expression: The addition and subtraction symbols separate this expression into three parts known as terms.

To identify each term, we put an addition symbol between each part and separate using parentheses:

So, the terms of this expression are: We say that is the first term, is the second term, and is the third term.

In this example, the subtraction symbol gave us the term . Whenever there is a subtraction symbol, the corresponding term will be negative.

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Example: Listing the Terms of a Linear Expression

List all terms of the expression

EXPLANATION

We put an addition symbol between each part and separate using parentheses:

So, the expression has three terms: , , and

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Practice: Listing the Terms of a Linear Expression

What are the terms of the expression $3a + 5?$

a	$3a$ and $5$
b	$3$ , $a$ , and $5$
c	$3$ and $5$
d	$a$ and $5$
e	$3$ and $a$

EXPLANATION

The expression $3a + 5$ has two terms: $3a$ and $5.$

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Practice: Listing the Terms of a Linear Expression

What are the terms of the expression $-12g - 7h + 3?$

a	$-12g$ , $-7h$ , and $3$
b	$-12g$ , $7h$ , and $3$
c	$g$ and $h$
d	$12g$ , $7h$ , and $3$
e	$12$ , $7$ , and $3$

EXPLANATION

We put an addition symbol $(+)$ between each part and separate using parentheses: $(-12g) + (-7h) + (3)$ So, the expression has three terms: $-12g$ , $-7h$ , and $3.$

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Practice: Listing the Terms of a Linear Expression

What are the terms of the expression $a - b -c +d?$

a	$-a$ , $b$ , $c$ , and $d$
b	$a$ , $b$ , $c$ , and $-d$
c	$-a$ , $-b$ , $c$ , and $d$
d	$a$ , $b$ , $c$ , and $d$
e	$a$ , $-b$ , $-c$ , and $d$

EXPLANATION

We put an addition symbol $(+)$ between each part and separate using parentheses: $(a)+( - b)+(- c) + (d)$ So, the expression has four terms: $a$ , $-b$ , $-c$ , and $d.$

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Example: Identifying Particular Terms of an Expression

What is the fourth term of the expression

EXPLANATION

We put an addition symbol between each part and separate using parentheses: Counting off the terms:

the first term is
the second term is
the third term is and
the fourth term is

In particular, we are interested in the fourth term. The fourth term is

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Practice: Identifying Particular Terms of an Expression

What's the first term of the expression $2x+y+z?$

a	$y$
b	$2x$
c	$z$
d	$x$
e	$-x$

EXPLANATION

We put an addition symbol $(+)$ between each part and separate using parentheses:

$(2x)+(y)+(z)$ Counting off the terms:

the first term is $2x,$
the second term is $y,$ and
the third term is $z.$

In particular, we are interested in the first term. The first term is $2x.$

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Practice: Identifying Particular Terms of an Expression

What is the fourth term of the expression $5 a -\dfrac 4 3 b - 12c+\dfrac 1 2 d?$

a	$-12c$
b	$-\dfrac 1 2 d$
c	$-\dfrac 4 3 b$
d	$\sqrt 5 a$
e	$\dfrac 1 2 d$

EXPLANATION

We put an addition symbol $(+)$ between each part and separate using parentheses: $( 5 a)+\left(-\dfrac 4 3 b\right)+( - 12c)+\left(\dfrac 1 2 d\right)$ Counting off the terms:

the first term is $5 a,$
the second term is $-\dfrac 4 3 b,$
the third term is $- 12c,$ and
the fourth term is $\dfrac 1 2 d.$

In particular, we are interested in the fourth term. The fourth term is $\dfrac 1 2 d.$

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Practice: Identifying Particular Terms of an Expression

What is the third term of the expression $4r+3s-t?$

a	$s$
b	$3s$
c	$t$
d	$r$
e	$-t$

EXPLANATION

We put an addition symbol $(+)$ between each part and separate using parentheses: $(4r) + (3s) + (-t)$ Counting off the terms:

the first term is $4r,$
the second term is $3s,$ and
the third term is $-t.$

In particular, we are interested in the third term. The third term is $-t.$

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Equivalent Expressions

When two algebraic expressions contain exactly the same terms, they are said to be equivalent.

For example, the expression is equivalent to because they both contain the exact same terms. Both expressions are also equivalent to

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Example: Equivalent Expressions

Write down an expression that is equivalent to

EXPLANATION

The expression has three terms: and

It doesn't matter in what order we write the terms. For example, one equivalent expression is:

Another equivalent expression is:

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Practice: Equivalent Expressions

Which of the following is equivalent to the expression $\dfrac13 b+2a?$

a	$2a+\dfrac13 b$
b	$-2a-\dfrac13 b$
c	$-\dfrac13a-2 b$
d	$\dfrac12a+3 b$
e	$\dfrac13a+2 b$

EXPLANATION

The expression $\dfrac13 b+2a$ has two terms: $\dfrac13 b$ and $2a.$

The order in which we combine the terms doesn't matter. Therefore, we can also write this expression as $2a+\dfrac13 b.$

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Practice: Equivalent Expressions

Which of the following is equivalent to the expression $3a-2b-c?$

a	$3a+2b-c$
b	$3a-2b+c$
c	$a-b-c$
d	$-c-3a+2b$
e	$-2b-c+3a$

EXPLANATION

The expression $3a-2b-c$ has three terms: $3a$ , $-2b$ , and $-c.$

The order in which we combine the terms doesn't matter. Therefore, we can also write this expression as $-2b-c+3a.$

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Practice: Equivalent Expressions

Which of the following is equivalent to the expression $6z+\dfrac{3}{2}x+2y+\dfrac{5}{2}?$

a	$2y+4+6z+x$
b	$6y+\dfrac{3}{2}+2z+\dfrac{5}{2}x$
c	$z+4x+2y$
d	$y+\dfrac{5}{2}+z+\dfrac{3}{2}x$
e	$2y+\dfrac{5}{2}+6z+\dfrac{3}{2}x$

EXPLANATION

The expression $6z+\dfrac{3}{2}x+2y+\dfrac{5}{2}$ has four terms: $6z$ , $\dfrac{3}{2}x$ , $2y$ and $\dfrac{5}{2}.$

The order in which we combine the terms doesn't matter. Therefore, we can also write this expression as $2y+\dfrac{5}{2}+6z+\dfrac{3}{2}x.$

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