Home / United States / Math Classes / 6th Grade Math / Writing Algebraic Expressions

Algebra is a branch of mathematics that deals with numbers, symbols, and mathematical operators. We will learn the steps involved in framing an algebraic equation that is relevant to a particular situation. The main aim of solving an algebraic expression is to obtain the value of an unknown variable....Read MoreRead Less

Algebra is a branch of mathematics that deals with symbols and the operations that can be performed on them. These symbols are referred to as variables because they do not have any fixed value.

Variables are entities that represent an unknown quantity or object by means of an alphabet.

Example: Let us consider there are two animals: a dog and a cat. So, the variable “a” represents the dog and the variable “b” represents the cat.

A number occurring in an algebraic expression that is “**invariable or unchanging” **is called a constant.

Examples of constants are 2, 8, 7, -5, -1, and so on.

Operators are mathematical symbols that emphasize a particular action on a set of constants or variables or both, such as addition, subtraction, division, and multiplication. Operands, on the other hand, can be defined as a number or value upon which the mathematical operation will be done.

In algebra, we use basic mathematical operations such as addition, subtraction, division, and multiplication using PEMDAS, which is a predefined rule to carry out various operations.

An algebraic expression is formed by using integers, constants, variables, and arithmetic operations. Variables and constants can be combined in an algebraic expression through mathematical operations.

For example, \(4\times~a=4a\) is an algebraic expression, where 4 is a constant, a is a variable, and the mathematical operation performed is multiplication.

The exponent of a number or a variable indicates the number of times it has been multiplied in the expression.

**For example, the expression:** \(2\cdot 1\cdot a\cdot a\cdot a\)

We have \(2\cdot 1\cdot a\cdot a\cdot a\) as an expression and here ‘a’ is used as a factor 3 times or multiplied 3 times, so its exponent is 3.

Hence, \(2\cdot 1~a^3\)

A numerical expression is a mathematical statement that is made up entirely of numbers and one or more operation symbols only.

For example, 3 + 4 – 1.

However, a + 9 – 1 + 6 = 11 is not a numerical expression, because of the variable ‘a’ in the expression.

**Write each phrase as a numerical expression.**

**Phrase: **The sum of 43 and 32.

**Expression: **43 + 32, because the term ‘sum’ refers to the addition of both the numbers.

**Phrase: **The product of 4 and 3.

**Expression: \(4\times~3\)**, because the term ‘product’ refers to multiplying both the numbers.

**Write each phrase as an expression.**

**Phrase: **A number m minus 123

**Expression: **m – 123

The term ‘minus’ refers to subtraction.

**Phrase: **The quotient of 5 and a number e.

**Expression: **5 ÷ e

The term ‘quotient’ refers to division.

**Example 1: **John has 25 toys more than twice of what you have. The number of toys you have is m. Write an expression for the total number of toys John has.

**Solution: **

The phrase ‘more than’ means addition.

The word ‘twice’ indicates multiplication by 2. Hence,

\(2\times~m+25\).

\(2m+25\)

The number of toys John has is \(2m+25\).

**Example 2: **Rose has 20 chocolates more than thrice of what Linda has. The number of chocolates Linda has is c. Write an expression for the total number of chocolates Rose has.

**Solution: **

The phrase ‘more than’ means addition.

The word ‘thrice’ indicates the multiplication of the chocolates by 3.

\(3\times~c+20\).

\(3c+20\)

The number of chocolates Rose has is \(3c+20\).

**Example 3: **Rewrite \((15\bullet~15\bullet~a\bullet~a)\) in the simplest form using exponents.

**Solution:**

Here we see that 15 and a are multiplied twice. Hence,

\(15\bullet~15\bullet~a\bullet~a=15^2a^2\)

**Example 4: **Evaluate: \((4y)^4\) if \(y=2\).

**Solution:**

We have \((4y)^4\) where \(y=2\)

\(=(4\bullet~2)^4\) Substitute 2 for the variable \(y\)

\(=(8)^4\) Evaluate \(8^4\)

= 4096

**Example 5: **You decide to plant a 20-inch-tall tree. It grows 5 inches in height every year. Make a formula to represent the height (in inches) after t years. After 10 years, what is the height?

**Solution: **Make a table that shows the tree’s height each year for the first few years. When \(t=10\), use the results to write an expression and evaluate it.

As the height grows, add 5 every year, as shown in the table.

Year (t) | Height (Inches) |
---|---|

0 | 20 + 5 (0) = 20 |

1 | 20 + 5 (1) = 25 |

2 | 20 + 5 (2) = 30 |

3 | 20 + 5 (3) = 35 |

4 | 20 + 5 (4) = 40 |

Now, we can form an expression using the table above for the height of the tree in terms of ‘t years’ = 20 + 5 (t).

As we have to find the height of the tree after 10 years, t = 10

Height = 20 + 5 (10) Multiply the height by the number of years

= 20 + 50 Simplify

= 70

So, the height of the tree after 10 years will be 70 inches.

Frequently Asked Questions

An algebraic expression is a mathematical phrase that uses the operational symbols (+, -, × and ÷) to combine variables and constants. The equal (=) sign is absent from an algebraic symbol. Algebraic expressions include, for example, (10x + 63) and (5x – 3).

We can represent the product in the following ways:

- 3.n
- 3 n
- 3 (n)