Review Point Task Math 2 11 Simplifying Expressions

2.three: Evaluate, Simplify, and Translate Expressions (Part 1)

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Skills to Develop

Evaluate algebraic expressions
Identify terms, coefficients, and like terms
Simplify expressions by combining similar terms
Translate word phrases to algebraic expressions

Be prepared!

Earlier you get started, take this readiness quiz.

Is $north ÷ five$ an expression or an equation? If you missed this problem, review Example 2.1.iv.
Simplify $4^5$. If yous missed this problem, review Example ii.ane.6.
Simplify $i + 8 • 9$. If you missed this problem, review Example ii.1.8.

Evaluate Algebraic Expressions

In the terminal section, we simplified expressions using the order of operations. In this section, nosotros'll evaluate expressions—once more post-obit the order of operations.

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and so simplify the expression using the club of operations.

Instance $\PageIndex{i}$: evaluate

Evaluate $10 + 7$ when

$x = three$
$10 = 12$

Solution

To evaluate, substitute $iii$ for $x$ in the expression, and and then simplify.

	$x + 7$
Substitute.	$\textcolor{red}{three} + 7$
Add.	$10$

When $x = 3$, the expression $x + 7$ has a value of $10$.

To evaluate, substitute $12$ for $10$ in the expression, and and then simplify.

	$ten + seven$
Substitute.	$\textcolor{ruby-red}{12} + 7$
Add.	$19$

When $ten = 12$, the expression $10 + vii$ has a value of $19$. Notice that we got different results for parts (a) and (b) fifty-fifty though nosotros started with the same expression. This is considering the values used for $10$ were unlike. When we evaluate an expression, the value varies depending on the value used for the variable.

do $\PageIndex{one}$

Evaluate: $y + iv$ when

$y = 6$
$y = 15$

Answer a: $10$
Answer b: $nineteen$

exercise $\PageIndex{2}$

Evaluate: $a − v$ when

$a = 9$
$a = 17$

Answer a: $four$
Respond b: $12$

Example $\PageIndex{2}$

Evaluate $9x − 2$, when

$x = five$
$10 = one$

Solution

Remember $ab$ means $a$ times $b$, and so $9x$ means $9$ times $x$.

To evaluate the expression when $ten = five$, nosotros substitute $5$ for $x$, then simplify.

	$9x - two$
Substitute $\textcolor{red}{5}$ for ten.	$9 \cdot \textcolor{red}{5} - 2$
Multiply.	$45 - two$
Subtract.	$43$

To evaluate the expression when $10 = ane$, we substitute $1$ for $x$, and so simplify.

	$9x - 2$
Substitute $\textcolor{cherry-red}{1}$ for x.	$ix \cdot \textcolor{crimson}{1} - 2$
Multiply.	$nine - 2$
Subtract.	$7$

Notice that in part (a) that nosotros wrote $9 • 5$ and in part (b) we wrote $ix(1)$. Both the dot and the parentheses tell the states to multiply.

exercise $\PageIndex{3}$

Evaluate: $8x − 3$, when

$10 = ii$
$10 = 1$

Answer a: $xiii$
Reply b: $5$

exercise $\PageIndex{four}$

Evaluate: $4y − 4$, when

$y = 3$
$y = 5$

Answer a: $viii$
Reply b: $xvi$

Example $\PageIndex{3}$: evaluate

Evaluate $x^ii$ when $x = 10$.

Solution

Nosotros substitute $10$ for $10$, and and so simplify the expression.

	$ten^{2}$
Substitute $\textcolor{ruby}{x}$ for 10.	$\textcolor{red}{ten}^{2}$
Utilize the definition of exponent.	$x \cdot 10$
Multiply	$100$

When $10 = 10$, the expression $x^ii$has a value of $100$.

practice $\PageIndex{v}$

Evaluate: $ten^2$ when $10 = 8$.

Answer: $64$

practice $\PageIndex{6}$

Evaluate: $10^three$when $x = half-dozen$.

Answer: $216$

Example $\PageIndex{4}$: evaluate

Evaluate $2^10$ when $x = 5$.

Solution

In this expression, the variable is an exponent.

	$two^{x}$
Substitute $\textcolor{cherry}{5}$ for x.	$2^{\textcolor{red}{v}}$
Use the definition of exponent.	$ii \cdot two \cdot 2 \cdot 2 \cdot 2$
Multiply	$32$

When $10 = 5$, the expression $ii^x$ has a value of $32$.

exercise $\PageIndex{seven}$

Evaluate: $2^10$ when $x = 6$.

Answer: $64$

practise $\PageIndex{8}$

Evaluate: $3^x$ when $x = iv$.

Answer: $81$

Example $\PageIndex{5}$: evaluate

Evaluate $3x + 4y − 6$ when $ten = ten$ and $y = ii$.

Solution

This expression contains two variables, so we must make two substitutions.

	$3x + 4y − 6$
Substitute $\textcolor{red}{x}$ for x and $\textcolor{blue}{ii}$ for y.	$3(\textcolor{scarlet}{10}) + iv(\textcolor{blue}{two}) − six$
Multiply.	$30 + eight - 6$
Add together and subtract left to right.	$32$

When $x = 10$ and $y = 2$, the expression $3x + 4y − 6$ has a value of $32$.

practise $\PageIndex{9}$

Evaluate: $2x + 5y − iv$ when $ten = eleven$ and $y = iii$

Reply: $33$

exercise $\PageIndex{10}$

Evaluate: $5x − 2y − 9$ when $x = vii$ and $y = 8$

Reply: $ten$

Example $\PageIndex{6}$: evaluate

Evaluate $2x^2 + 3x + eight$ when $x = 4$.

Solution

We need to be careful when an expression has a variable with an exponent. In this expression, $2x^2$ means $2 • x • 10$ and is different from the expression $(2x)^2$, which means $2x • 2x$.

	$2x^{2} + 3x + 8$
Substitute $\textcolor{cherry-red}{four}$ for each ten.	$2(\textcolor{ruddy}{4})^{two} + 3(\textcolor{red}{four}) + eight$
Simplify ivⁱⁱ.	$2(16) + iii(4) + 8$
Multiply.	$32 + 12 + eight$
Add.	$52$

practice $\PageIndex{eleven}$

Evaluate: $3x^ii + 4x + 1$ when $ten = 3$.

Reply: $40$

practice $\PageIndex{12}$

Evaluate: $6x^2 − 4x − vii$ when $x = 2$.

Answer: $nine$

Identify Terms, Coefficients, and Like Terms

Algebraic expressions are fabricated up of terms. A term is a abiding or the product of a constant and one or more variables. Some examples of terms are $7$, $y$, $5x^2$, $9a$, and $13xy$.

The constant that multiplies the variable(south) in a term is called the coefficient. We tin can think of the coefficient as the number in front of the variable. The coefficient of the term $3x$ is $3$. When we write $ten$, the coefficient is $1$, since $x = i • 10$. Table $\PageIndex{1}$ gives the coefficients for each of the terms in the left column.

Table $\PageIndex{i}$
Term	Coefficient
7	seven
9a	9
y	one
5xⁱⁱ	5

An algebraic expression may consist of i or more than terms added or subtracted. In this chapter, we volition only work with terms that are added together. Tabular array $\PageIndex{two}$ gives some examples of algebraic expressions with diverse numbers of terms. Discover that nosotros include the performance before a term with it.

Table $\PageIndex{2}$
Expression	Terms
7	vii
y	y
x + 7	x, 7
2x + 7y + 4	2x, 7y, 4
3x²+ 4x^two + 5y + 3	3x^ii, 4x², 5y, three

Example $\PageIndex{7}$:

Identify each term in the expression $9b + 15x^two + a + 6$. So identify the coefficient of each term.

Solution

The expression has four terms. They are $9b$, $15x^2$, $a$, and $half dozen$.

The coefficient of $9b$ is $ix$.

The coefficient of $15x^2$ is $15$.

Think that if no number is written before a variable, the coefficient is $1$. So the coefficient of a is $ane$.

The coefficient of a constant is the abiding, so the coefficient of $6$ is $6$.

exercise $\PageIndex{13}$

Place all terms in the given expression, and their coefficients: $4x + 3b + ii$

Reply: The terms are $4x, 3b,$ and $2$. The coefficients are $4, 3,$ and $two$.

exercise $\PageIndex{14}$

Place all terms in the given expression, and their coefficients: $9a + 13a^2 + a^3$

Answer: The terms are $9a, 13a^2,$ and $a^3$, The coefficients are $nine, 13,$ and $1$.

Some terms share common traits. Look at the post-obit terms. Which ones seem to accept traits in mutual?

$5x, 7, n^{2}, 4, 3x, 9n^{ii}$

Which of these terms are like terms?

The terms $7$ and $4$ are both constant terms.
The terms $5x$ and $3x$ are both terms with $10$.
The terms $n^2$ and $9n^two$ both take $n^2$.

Terms are called similar terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms $5x, 7, north^2, 4, 3x, 9n^two, 7$ and $4$ are like terms, $5x$ and $3x$ are like terms, and $n^2$ and $9n^2$ are like terms.

Definition: Similar terms

Terms that are either constants or have the aforementioned variables with the same exponents are like terms.

Case $\PageIndex{8}$: identify

Place the similar terms:

$y^3, 7x^2, 14, 23, 4y^3, 9x, 5x^two$
$4x^2 + 2x + 5x^2 + 6x + 40x + 8xy$

Solution

$y^iii, 7x^2, 14, 23, 4y^3, 9x, 5x^2$

Look at the variables and exponents. The expression contains $y^3, x^2, x$, and constants. The terms $y^iii$ and $4y^3$ are like terms because they both take $y^iii$. The terms $7x^2$ and $5x^2$ are like terms because they both take $x^2$. The terms $xiv$ and $23$ are like terms considering they are both constants. The term $9x$ does not take any like terms in this listing since no other terms have the variable $x$ raised to the power of $1$.

$4x^ii + 2x + 5x^2 + 6x + 40x + 8xy$

Look at the variables and exponents. The expression contains the terms $4x^ii, 2x, 5x^two, 6x, 40x$, and $8xy$ The terms $4x^2$ and $5x^ii$ are like terms because they both have $ten^2$. The terms $2x, 6x$, and $40x$ are like terms because they all have $ten$. The term $8xy$ has no like terms in the given expression because no other terms contain the two variables $xy$.

exercise $\PageIndex{15}$

Identify the like terms in the listing or the expression: $nine, 2x^3, y^2, 8x^3, 15, 9y, 11y^2$

Answer: $ix, xv$; $2x^3$ and $8x^three$, $y^2$, and $11y^2$

exercise $\PageIndex{16}$

Place the like terms in the listing or the expression: $4x^iii + 8x^2 + nineteen + 3x^2 + 24 + 6x^3$

Answer: $4x^3$ and $6x^iii$; $8x^ii$ and $3x^2$; $19$ and $24$

Simplify Expressions past Combining Similar Terms

We can simplify an expression by combining the like terms. What do yous think $3x + 6x$ would simplify to? If you lot thought $9x$, y'all would exist right!

We tin can see why this works by writing both terms equally addition bug.

$CNX_BMath_Figure_02_02_001_img.jpg$

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 instance, $three$ oranges plus $half-dozen$ oranges is $9$ oranges. We will discuss the mathematical properties backside this later on.

The expression $3x + 6x$ has only 2 terms. When an expression contains more terms, it may exist helpful to rearrange the terms so that similar terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. Then we could rearrange the post-obit expression before combining like terms.

$CNX_BMath_Figure_02_02_015_img.jpg$

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

HOW TO: COMBINE LIKE TERMS

Pace one. Identify similar terms.

Footstep 2. Rearrange the expression so similar terms are together.

Step 3. Add together the coefficients of the like terms.

Example $\PageIndex{ix}$: simplify

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

Solution

	$3x + seven + 4x + 5$
Identify the like terms	$\textcolor{carmine}{3x} + \textcolor{blue}{seven} + \textcolor{cherry}{4x} + \textcolor{blue}{five}$
Rearrange the expression, so the like terms are together.	$\textcolor{cherry-red}{3x} + \textcolor{red}{4x} + \textcolor{bluish}{7} + \textcolor{blueish}{5}$
Add the coefficients of the similar terms.	$\textcolor{carmine}{7x} + \textcolor{blue}{12}$
The original expression is simplified to...	$7x + 12$

do $\PageIndex{17}$

Simplify: $7x + 9 + 9x + 8$

Answer: $16x+17$

exercise $\PageIndex{18}$

Simplify: $5y + 2 + 8y + 4y + 5$

Answer: $17y+7$

Instance $\PageIndex{10}$: simplify

Simplify the expression: $7x^ii + 8x + 10^2 + 4x$.

Solution

	$7x^{2} + 8x + x^{two} + 4x$
Identify the like terms.	$\textcolor{cerise}{7x^{ii}} + \textcolor{bluish}{8x} + \textcolor{ruby-red}{x^{2}} + \textcolor{blueish}{4x}$
Rearrange the expression so like terms are together.	$\textcolor{red}{7x^{2}} + \textcolor{scarlet}{x^{2}} + \textcolor{blueish}{8x} + \textcolor{bluish}{4x}$
Add the coefficients of the similar terms.	$\textcolor{cherry-red}{8x^{2}} + \textcolor{blue}{12x}$

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

exercise $\PageIndex{19}$

Simplify: $3x^2 + 9x + x^2 + 5x$

Answer: $4x^2+14x$

exercise $\PageIndex{twenty}$

Simplify: $11y^2 + 8y + y^two + 7y$

Answer: $12y^2+15y$

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Artistic Commons Attribution License four.0 license.

carraffor1956.blogspot.com

Source: https://math.libretexts.org/Bookshelves/PreAlgebra/Book:_Prealgebra_%28OpenStax%29/02:_Introduction_to_the_Language_of_Algebra/2.03:_Evaluate_Simplify_and_Translate_Expressions_%28Part_1%29

	\(x + 7\)
Substitute.	\(\textcolor{red}{three} + 7\)
Add.	\(10\)

	\(ten + seven\)
Substitute.	\(\textcolor{ruby-red}{12} + 7\)
Add.	\(19\)

	\(9x - two\)
Substitute \(\textcolor{red}{5}\) for ten.	\(9 \cdot \textcolor{red}{5} - 2\)
Multiply.	\(45 - two\)
Subtract.	\(43\)

	\(9x - 2\)
Substitute \(\textcolor{cherry-red}{1}\) for x.	\(ix \cdot \textcolor{crimson}{1} - 2\)
Multiply.	\(nine - 2\)
Subtract.	\(7\)

	\(ten^{2}\)
Substitute \(\textcolor{ruby}{x}\) for 10.	\(\textcolor{red}{ten}^{2}\)
Utilize the definition of exponent.	\(x \cdot 10\)
Multiply	\(100\)

Review Point Task Math 2 11 Simplifying Expressions

2.three: Evaluate, Simplify, and Translate Expressions (Part 1)

Evaluate Algebraic Expressions

Identify Terms, Coefficients, and Like Terms

Simplify Expressions past Combining Similar Terms

Contributors and Attributions

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	\(two^{x}\)
Substitute \(\textcolor{cherry}{5}\) for x.	\(2^{\textcolor{red}{v}}\)
Use the definition of exponent.	\(ii \cdot two \cdot 2 \cdot 2 \cdot 2\)
Multiply	\(32\)

	\(3x + 4y − 6\)
Substitute \(\textcolor{red}{x}\) for x and \(\textcolor{blue}{ii}\) for y.	\(3(\textcolor{scarlet}{10}) + iv(\textcolor{blue}{two}) − six\)
Multiply.	\(30 + eight - 6\)
Add together and subtract left to right.	\(32\)

	\(2x^{2} + 3x + 8\)
Substitute \(\textcolor{cherry-red}{four}\) for each ten.	\(2(\textcolor{ruddy}{4})^{two} + 3(\textcolor{red}{four}) + eight\)
Simplify ivⁱⁱ.	\(2(16) + iii(4) + 8\)
Multiply.	\(32 + 12 + eight\)
Add.	\(52\)

	\(3x + seven + 4x + 5\)
Identify the like terms	\(\textcolor{carmine}{3x} + \textcolor{blue}{seven} + \textcolor{cherry}{4x} + \textcolor{blue}{five}\)
Rearrange the expression, so the like terms are together.	\(\textcolor{cherry-red}{3x} + \textcolor{red}{4x} + \textcolor{bluish}{7} + \textcolor{blueish}{5}\)
Add the coefficients of the similar terms.	\(\textcolor{carmine}{7x} + \textcolor{blue}{12}\)
The original expression is simplified to...	\(7x + 12\)

	\(7x^{2} + 8x + x^{two} + 4x\)
Identify the like terms.	\(\textcolor{cerise}{7x^{ii}} + \textcolor{bluish}{8x} + \textcolor{ruby-red}{x^{2}} + \textcolor{blueish}{4x}\)
Rearrange the expression so like terms are together.	\(\textcolor{red}{7x^{2}} + \textcolor{scarlet}{x^{2}} + \textcolor{blueish}{8x} + \textcolor{bluish}{4x}\)
Add the coefficients of the similar terms.	\(\textcolor{cherry-red}{8x^{2}} + \textcolor{blue}{12x}\)