2.2 Evaluate, simplify, and translate expressions (Page 3/9)

Prealgebra Page 3 / 9

We can see why this works by writing both terms as addition problems.

The image shows the expression 3 x plus 6 x. The 3 x represents x plus x plus x. The 6 x represents x plus x plus x plus x plus x plus x. The expression 3 x plus 6 x becomes x plus x plus x plus x plus x plus x plus x plus x plus x. This simplifies to a total of 9 x's or the term 9 x.

Add the coefficients and keep the same variable. It doesn’t matter what $x$ is. If you have $3$ of something and add $6$ more of the same thing, the result is $9$ of them. For example, $3$ oranges plus $6$ oranges is $9$ oranges. We will discuss the mathematical properties behind this later.

The expression $3 x + 6 x$ has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms.

The image shows the expression 3 x plus 4 y plus 2 x plus 6 y. The position of the middle terms, 4 y and 2 x, can be switched so that the expression becomes 3 x plus 2 x plus 4 y plus 6 y. Now the terms containing x are together and the terms containing y are together.

Now it is easier to see the like terms to be combined.

Combine like terms.

Identify like terms.
Rearrange the expression so like terms are together.
Add the coefficients of the like terms.

Simplify the expression: $3 x + 7 + 4 x + 5 .$

Solution


Identify the like terms.
Rearrange the expression, so the like terms are together.
Add the coefficients of the like terms.
The original expression is simplified to...

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Simplify:

$7 x + 9 + 9 x + 8$

16 x + 17

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Simplify:

$5 y + 2 + 8 y + 4 y + 5$

17 y + 7

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Simplify the expression: $7 x^{2} + 8 x + x^{2} + 4 x .$

Solution


Identify the like terms.
Rearrange the expression so like terms are together.
Add the coefficients of the like terms.

These are not like terms and cannot be combined. So $8 x^{2} + 12 x$ is in simplest form.

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Simplify:

$3 x^{2} + 9 x + x^{2} + 5 x$

4 x ² + 14 x

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Simplify:

$11 y^{2} + 8 y + y^{2} + 7 y$

12 y ² + 15 y

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Translate words to algebraic expressions

In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Now we’ll reverse the process and translate word phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. They are summarized in [link]

Operation	Phrase	Expression
Addition	$a plus b$ $\begin{array}{l} the sum of a and b \end{array}$ $a increased by b$ $b more than a$ $\begin{array}{l} the total of a and b \end{array}$ $b added to a$	$a + b$
Subtraction	$a minus b$ $the difference of a and b$ $b subtracted from a$ $a decreased by b$ $b less than a$	$a - b$
Multiplication	$a times b$ $the product of a and b$	$a \cdot b, a b, a (b), (a) (b)$
Division	$a divided by b$ $the quotient of a and b$ $the ratio of a and b$ $b divided into a$	$a \div b, a / b, \frac{a}{b}, b a$

Operation	Phrase	Expression
Addition	$a$ plus $b$ the sum of $a$ and $b$ $a$ increased by $b$ $b$ more than $a$ the total of $a$ and $b$ $b$ added to $a$	$a + b$
Subtraction	$a$ minus $b$ the difference of $a$ and $b$ $b$ subtracted from $a$ $a$ decreased by $b$ $b$ less than $a$	$a - b$
Multiplication	$a$ times $b$ the product of $a$ and $b$	$a \cdot b$ , $a b$ , $a (b)$ , $(a) (b)$
Division	$a$ divided by $b$ the quotient of $a$ and $b$ the ratio of $a$ and $b$ $b$ divided into $a$	$a \div b$ , $a / b$ , $\frac{a}{b}$ , $b a$

Look closely at these phrases using the four operations:

the sum of $a$ and $b$
the difference of $a$ and $b$
the product of $a$ and $b$
the quotient of $a$ and $b$

Each phrase tells you to operate on two numbers. Look for the words of and and to find the numbers.

Translate each word phrase into an algebraic expression:

ⓐ the difference of $20$ and $4$
ⓑ the quotient of $10 x$ and $3$

Solution

ⓐ The key word is difference , which tells us the operation is subtraction. Look for the words of and and to find the numbers to subtract.

$\begin{array}{l} the difference of 20 a n d 4 \\ 20 minus 4 \\ 20 - 4 \end{array}$

ⓑ The key word is quotient , which tells us the operation is division.

$\begin{array}{l} the quotient of 10 x and 3 \\ divide 10 x by 3 \\ 10 x \div 3 \end{array}$

This can also be written as $\begin{array}{l} 10 x / 3 or \frac{10 x}{3} \end{array}$

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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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