Algebraic Expressions

Learning Objectives

After completing this section, you should be able to:

1. Convert between written and symbolic algebraic expressions and equations.
2. Simplify and evaluate algebraic expressions.
3. Add and subtract algebraic expressions.
4. Multiply and divide algebraic expressions.

Algebraic Expressions and Equations

Xavier and Yasenia have the same birthday, but they were born in different years. This year Xavier is 20 years old and Yasenia is 23, so Yasenia is three years older than Xavier. When Xavier was 15, Yasenia was 18. When Xavier will be 33, Yasenia will be 36. No matter what Xavier’s age is, Yasenia’s age will always be 3 years more.

In the language of algebra, we say that Xavier's age and Yasenia's age are variable and the 3 is a constant. The ages change, or vary, so age is a variable. The 3 years between them always stays the same or has the same value, so the age difference is the constant. In algebra, letters of the alphabet are used to represent variables. The letters most often used for variables are \(x\), \(y\), \(z\), \(a\), \(b\), and \(c\). Suppose we call Xavier's age \(x\). Then we could use \(x+3\) to represent Yasenia's age, as shown in the table below.

Xavier’s Age	Yasenia’s Age
15	18
20	23
33	36
\(x\)	\(x+3\)

To write algebraically, we need some symbols as well as numbers and variables. The symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division are summarized in , along with words we use for the operations and the result.

Operation	Notation	Say:	The result is…
Addition	\(a+b\)	\(a\) plus \(b\)	The sum of \(a\) and \(b\)
Subtraction	\(a\) − \(b\)	\(a\) minus \(b\)	The difference of \(a\) and \(b\)
Multiplication	\(a\) • \(b\), (\(a\))(\(b\)), (\(a\))\(b\), \(a\)(\(b\)), \(\text{ab}\), \(\text{ba}\)	\(a\) times \(b\)	The product of \(a\) and \(b\)
Division	\(a\) ÷ \(b\), \(a\)/\(b\)	\(a\) divided by \(b\)	The quotient of \(a\) and \(b\)

We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.

• The sum of 5 and 3 means add 5 plus 3, which we write as \(5+3\).
• The difference of 9 and 2 means subtract 9 minus 2, which we write as \(9-2\).
• The product of 4 and 8 means multiply 4 times 8, which we can write as \(4•8\).
• The quotient of 20 and 5 means divide 20 by 5, which we can write as \(20\div 5\).

Translating from Words to Algebra

Try it.

Translate the following phrases from words into algebraic expressions.

1. The difference of 47 and 19
2. 72 divided by 9
3. The sum of \(m\) and \(n\)
4. 13 times 7

Solution

1. According to , these words could be translated as \(47-19\).
2. According to , these words could be translated as \(72\div 9\).
3. According to , these words could be translated as \(m+n\).
4. According to , these words could be translated as \((13)(7)\).

Condensed: the full section is in OpenStax Contemporary Mathematics.

Simplifying and Evaluating Algebraic Expressions

To simplify an expression means to do all the math possible. For example, to simplify \(4•2+1\) we would first multiply \(4•2\) to get 8 and then add 1 to get 9. We have introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values. Consider \(2+7•3\). Do you add first or multiply first? Do you get different answers?

Add first: \(9•3=27\)

Multiply first: \(2+21=23\)

Which one is correct?

Early on, mathematicians realized the need to establish some guidelines when performing arithmetic operations to ensure that everyone would get the same answer. Those guidelines are called the order of operations and are listed in the table below.

Step 1: Parentheses and Other Grouping Symbols	Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
Step 2: Exponents	Simplify all expressions with exponents.
Step 3: Multiplication and Division	Perform all multiplication and division in order from left to right. These operations have equal priority.
Step 4: Addition and Subtraction	Perform all addition and subtraction in order from left to right. These operations have equal priority.

Making a Numerical Equation True Using the Order of Operations

Try it.

Use parentheses to make the following statements true.

1. \(17-10+3=10\)
2. \(2•26-7=38\)
3. \(8+12\div 5-3=14\)
4. \(5+{2}^{3}•7=91\)

Solution

1. Add the parentheses around the \(17-10\). Then you have \((17-10)+3=7+3=10\).
2. Add the parentheses around the \(26-7\). Then you have \(2•(26-7)=2•19=38\).
3. Add the parentheses around the \(5-3\). Then you have \(8+12\div (5-3)=8+12\div 2=8+6=14\).
4. Add the parentheses around the \(5+{2}^{3}\). Then you have \((5+{2}^{3})•7=(5+8)•7=13•7=91\).

In the last example, we simplified expressions using the order of operations. Now we'll evaluate some expressions, again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

Condensed: the full section is in OpenStax Contemporary Mathematics.

Operations of Algebraic Expressions

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Examples of terms are 7, \(y\), 5\({x}^{2}\), 9\(a\), and \({b}^{5}\). The constant that multiplies the variable is called the coefficient. Think of the coefficient as the number in front of the variable. Consider the algebraic expressions 5\({x}^{2}\), which has a coefficient of 5, and 9\(a\), which has a coefficient of 9. If there is no number listed in front of the variable, then the coefficient is 1 since \(x=1•x\).

Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they are like terms. If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.

Adding Algebraic Expressions

Try it.

Add \(({x}^{2}+4x-9)+(3{x}^{2}-x+12)\).

Solution

Step 1: Add the terms in any order and get the same result (think: \(2+3=3+2\) ) and drop the parentheses: \[{x}^{2}+4x-9+3{x}^{2}-x+12\]

Step 2: Group like terms together: \[{x}^{2}+3{x}^{2}+4x-x-9+12\]

Step 3: Combine the like terms:

\[4{x}^{2}+3x+3\]

Subtracting Algebraic Expressions

Try it.

Subtract \((5{x}^{2}+4x-9)-(3{x}^{2}-x+12)\).

Solution

Step 1: Distribute the negative inside the parentheses (think: \(2-(3-4)=2-3+4=-1+4=3\), which is the correct answer). You cannot just drop the parentheses (for example, \(2-3-4=-1-4=-5\), which is not correct as we have already verified the answer is 3): \[5{x}^{2}+4x-9-3{x}^{2}+x-12\]

Step 2: Group like terms together: \[5{x}^{2}-3{x}^{2}+4x+x-9-12\]

Step 3: Combine the like terms:

\[2{x}^{2}+x-21\]

Before looking at multiplying algebraic expressions we look at the Distributive Property, which says that to multiply a sum, first you multiply each term in the sum and then you add the products. For example, \(5(4+3)=5(4)+5(3)=20+15=35\) can also be solved as \(5(4+3)=5(7)=35\). If we use a variable, then \(5(x+3)=5x+15\).

We can extended this example to \((5+2)(4+3)=(5)(4)+(5)(3)+(2)(4)+(2)(3)=20+15+8+6=49\), which can also be solved as \((5+2)(4+3)=(7)(7)=49\). If we use variables, then \((x+5)(x+4)=(x)(x)+(x)(4)+(5)(x)+(5)(4)={x}^{2}+4x+5x+20={x}^{2}+9x+20\).

Simplifying an Expression Using the Order of Operations

Try it.

Simplify each expression.

1. \((x-3)5\)
2. \((-3)(x+y-2)\)
3. \({5}^{2}(7+3)(x)\)
4. \(4+x•5\)
5. \((4+x)•5\)

Solution

1. \(5x-3•5=5x-15\)
2. \((-3)•x+(-3)•y-(-3)•2=-3x-3y+6\)
3. \(25(7+3)(x)=25(10)(x)=250x\)
4. \(4+5x\)
5. \((4)•(5)+(x)•(5)=20+5x\)

Dividing Algebraic Expressions

Try it.

Divide \((8{x}^{2}+4x-16)\div (4x)\).

Solution

Divide EACH term by 4\(x\):

\[(8{x}^{2}\div 4x)+(4x\div 4x)-(16\div 4x)=2x+1-\frac{4}{x}\]

Condensed: the full section is in OpenStax Contemporary Mathematics.

Key Concepts

• Algebra is useful because it allows us to understand many situations in real life by modeling them with expressions.
• Algebraic expressions are the building blocks of algebra. From algebraic expressions we can create algebraic equations.
• Algebraic expressions are often simplified and evaluated using the four arithmetic operations.

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1. Translate the following algebraic expressions from algebra into words.

   1. \(12+14\)
   2. \((30)(5)\)
   3. \(64\div 8\)
   4. \(x-y\)
   끝내주는 걸 힌트

   답을 드러내세요

   1. According to , this could be translated as 12 plus 14 OR the sum of 12 and 14.
   2. According to , this could be translated as 30 times 5 OR the product of 30 and 5.
   3. According to , this could be translated as 64 divided by 8 OR the quotient of 64 and 8.
   4. According to , this could be translated as \(x\) minus \(y\) OR the difference of \(x\) and \(y.\)

2. Translate the following phrases from words into algebraic expressions.

   1. The difference of 47 and 19
   2. 72 divided by 9
   3. The sum of \(m\) and \(n\)
   4. 13 times 7
   끝내주는 걸 힌트

   답을 드러내세요

   1. According to , these words could be translated as \(47-19\).
   2. According to , these words could be translated as \(72\div 9\).
   3. According to , these words could be translated as \(m+n\).
   4. According to , these words could be translated as \((13)(7)\).

3. Translate the following phrases from words into algebraic expressions.

   1. Seven more than a number \(n\).
   2. A number \(n\) times itself.
   3. Six times a number \(n\), plus two more.
   4. The cost of postage is a flat rate of 10 cents for every parcel, plus 34 cents per ounce \(x\).
   끝내주는 걸 힌트

   답을 드러내세요

   1. \(n+7\)
   2. \(n•n\) or \({n}^{2}\)
   3. \(6n+2\)
   4. \(10+34x\)

4. Translate the following sentences from words into algebraic equations.

   1. Two times \(x\) is 6.
   2. \(n\) plus 2 is equal to \(n\) times 3.
   3. The quotient of 35 and 7 is 5.
   4. Sixty-seven minus \(x\) is 56.
   끝내주는 걸 힌트

   답을 드러내세요

   1. \(2x=6\)
   2. \(n+2=3n\)
   3. \(35\div 7=5\)
   4. \(67-x=56\)

5. Use parentheses to make the following statements true.

   1. \(17-10+3=10\)
   2. \(2•26-7=38\)
   3. \(8+12\div 5-3=14\)
   4. \(5+{2}^{3}•7=91\)
   끝내주는 걸 힌트

   답을 드러내세요

   1. Add the parentheses around the \(17-10\). Then you have \((17-10)+3=7+3=10\).
   2. Add the parentheses around the \(26-7\). Then you have \(2•(26-7)=2•19=38\).
   3. Add the parentheses around the \(5-3\). Then you have \(8+12\div (5-3)=8+12\div 2=8+6=14\).
   4. Add the parentheses around the \(5+{2}^{3}\). Then you have \((5+{2}^{3})•7=(5+8)•7=13•7=91\).
6. 1. Evaluate \(3x+5\) when \(x=2\).
   2. Evaluate \({x}^{2}+3x+1\) when \(x=2\).
   끝내주는 걸 힌트

   답을 드러내세요

   1. To evaluate, let \(x=2\) in the expression, and then simplify: \(3(2)+5=6+5=11\).
   2. To evaluate, let \(x=2\) in the expression, and then simplify: \({2}^{2}+3(2)+1=4+6+1=11\).

7. Add \(({x}^{2}+4x-9)+(3{x}^{2}-x+12)\).

   끝내주는 걸 힌트

   답을 드러내세요

   Step 1: Add the terms in any order and get the same result (think: \(2+3=3+2\) ) and drop the parentheses: \[{x}^{2}+4x-9+3{x}^{2}-x+12\]

   Step 2: Group like terms together: \[{x}^{2}+3{x}^{2}+4x-x-9+12\]

   Step 3: Combine the like terms:

   \[4{x}^{2}+3x+3\]

8. Subtract \((5{x}^{2}+4x-9)-(3{x}^{2}-x+12)\).

   끝내주는 걸 힌트

   답을 드러내세요

   Step 1: Distribute the negative inside the parentheses (think: \(2-(3-4)=2-3+4=-1+4=3\), which is the correct answer). You cannot just drop the parentheses (for example, \(2-3-4=-1-4=-5\), which is not correct as we have already verified the answer is 3): \[5{x}^{2}+4x-9-3{x}^{2}+x-12\]

   Step 2: Group like terms together: \[5{x}^{2}-3{x}^{2}+4x+x-9-12\]

   Step 3: Combine the like terms:

   \[2{x}^{2}+x-21\]

9. Simplify each expression.

   1. \((x-3)5\)
   2. \((-3)(x+y-2)\)
   3. \({5}^{2}(7+3)(x)\)
   4. \(4+x•5\)
   5. \((4+x)•5\)
   끝내주는 걸 힌트

   답을 드러내세요

   1. \(5x-3•5=5x-15\)
   2. \((-3)•x+(-3)•y-(-3)•2=-3x-3y+6\)
   3. \(25(7+3)(x)=25(10)(x)=250x\)
   4. \(4+5x\)
   5. \((4)•(5)+(x)•(5)=20+5x\)

10. Multiply \((4x-9)(x+2)\).

    끝내주는 걸 힌트

    답을 드러내세요

    Step 1: Use the Distributive Property: \[(4x)(x)+(4x)(2)-(9)(x)-(9)(2)\]

    Step 2: Multiply: \[4{x}^{2}+8x-9x-18\]

    Step 3: Combine the like terms:

    \[4{x}^{2}-x-18\]

11. Divide \((8{x}^{2}+4x-16)\div (4x)\).

    끝내주는 걸 힌트

    답을 드러내세요

    Divide EACH term by 4\(x\):

    \[(8{x}^{2}\div 4x)+(4x\div 4x)-(16\div 4x)=2x+1-\frac{4}{x}\]

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\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\neqnot equal
The two sides are different.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.|x|absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.iimaginary unit
i² = −1.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

어떻게: Algebraic Expressions

1. Convert between written and symbolic algebraic expressions and equations.
2. Simplify and evaluate algebraic expressions.
3. Add and subtract algebraic expressions.
4. Multiply and divide algebraic expressions.
5. The
6. The
7. The
8. The

사람들이 묻는 질문

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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