__maths__>__Introduction to Algebra, Polynomials, and Identities__>__Algebra: Variables and Expressions__### Algebraic Expressions

In this page, algebraic expression is introduced as a collection of variables, numbers, and arithmetic operations between them representing a quantity. The terms and coefficients of algebraic expression is explained.

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As part of numerical arithmetic, students are introduced to numerical expressions. Let us just recall what we know already. Which of the following is an example of numerical expression?

- `2+2+84-2`
- `23 xx 123 + 8`
- `(32-23)+43xx2 - 4`
- all the above
- all the above

The answer is 'all the above'

What is a numerical expression?

- a collection of numbers and arithmetic operations between them forming a quantity
- a statement of a quantity with numbers and arithmetic operations
- both the above
- both the above

The answer is 'both the above'

Algebraic expressions are similar to numerical expressions. Let us see some examples to understand algebraic expressions.

How many squares are made in the figure?

- `3`
- `3`
- `1`

The answer is '`3`'. There are `3` squared made using crayons.

How many crayons are used in the figure?

- `3`
- `10`
- `10`

The answer is '`10`'. There are `3` squared made using `10` crayons

In this pattern of squares using crayons, if the number of squares in a row is `2`, how many crayons are required? Note: the figure shows `3` squares, but the question asks about `2` squares.

- `4`
- `7`
- `7`

The answer is '`7`'

In this pattern of squares using crayons, if the number of squares in a row is `10`, how many crayons are required? Note: the figure shows `3` squares, but the question asks about `10` squares.

- cannot figure this out
- `3xx10 + 1`
- `3xx10 + 1`

The answer is '`3xx10 + 1`'. This is an example of numerical expression.

In this pattern of squares using crayons, if the number of squares in a row is `x`, how many crayons are required? Note: the figure shows `3` squares, but the question asks about `x` squares.

- `3xx x + 1`
- `3xx x + 1`
- `4xx x`

The answer is '`3xx x + 1`'. This expression involves a variable `x`, and this is an example of algebraic expression.

The figure shows the how the formula is arrived at. For each of `3` columns, `3` crayons are used -- shown by shaded regions. There is an additional `1` on the left side. For `x` columns, there will be `x` shaded regions with `3x` crayons and `1` additional crayon.

Number of crayons required for `x` columns = `3x+1`

`3x+1` is an example of algebraic expression.

How many squares are made in the figure?

- `3`
- `6`
- `6`

The answer is '`6`'. There are `6` squares made using crayons.

How many crayons are used in the figure?

- `17`
- `17`
- `10`

The answer is '`17`'. There are `6` squared made using `17` crayons

In this pattern of squares using crayons, the figure shows `2` rows and `3` columns. If the number of rows and columns are changed to `4` and `5`, how many crayons are required?

- `4+5+2xx4xx5`
- `4+5+2xx4xx5`
- `3xx5+1`

The answer is '`4+5+2xx4xx5`'. Derivation of this formula is explained in the next page.

The figure shows the pattern with `2` rows and `3` columns.

• The crayons highlighted with blue background on the left equals number of rows `2`.

• The crayons highlighted with red background at the bottom equals number of columns `3`.

• The crayons highlighted with green and yellow backgrounds are repeated the number of squares `2xx2xx3`.

Total number of crayons `= 2+3+2xx2xx3 =17` `2+3+2xx2xx3` is an example of numerical expression. It evaluates to `17`.

Generalizing this for `x` rows and `y` columns.

• The crayons on the left equals number of rows `x`.

• The crayons at the bottom equals number of columns `y`.

• The crayons on squares are repeated the number of squares `2xx x xx y`.

The total number of crayons `=x+y+2xy`.

The result `x+y+2xy` is an example of an algebraic expression. The following are noted for this expression.

• there are `3` terms:, `x, y,` and `2xy`

• there are `2` variables : `x`, and `y`

• the constant in the term `2xy`, is `2` and it is called coefficient of that term.

Which of the following is a meaning for the word 'expression'?

- a collection that jointly represent a quantity
- a collection that jointly represent a quantity
- something that is moving very fast

The answer is 'a collection that jointly represent a quantity'.

Which of the following is a meaning for the word 'term'?

- one smaller part of quantity
- one smaller part of quantity
- year by year

The answer is 'one smaller part of quantity'.

Which of the following is a meaning for the word 'coefficient'?

- being together to produce a result
- being together to produce a result
- very efficient

The answer is 'being together to produce a result'. coefficient is co (together) + effect (root word for producing a result).

What is the word or phrase used to refer a quantity that is given by collection of variables, numbers, and arithmetic between them?

- Pronunciation : Say the answer once

Spelling: Write the answer once

The answer is 'algebraic expression'.

What is the word or phrase used to refer to parts of an algebraic expression?

- Pronunciation : Say the answer once

Spelling: Write the answer once

The answer is 'algebraic term or term'.

What is the word used to refer the number part in a term?

- Pronunciation : Say the answer once

Spelling: Write the answer once

The answer is 'coefficient of the term'.

**Algebraic Expression :** A quantity represented by collection of variables, numbers, and arithmetic operations between them is an algebraic expression. **Algebraic Term :** A part of an algebraic expression which is added to the rest of the expression in forming the expression.**Coefficient :** The constant part of a term is called the coefficient of the term.

*Solved Exercise Problem: *

A rectangle is made by joining two squares of side `x`. What is the outer perimeter of the rectangle?

- `4 x`
- `6 x`
- `6 x`

The answer is '`6 x`'

*Solved Exercise Problem: *

A person covers `5000` steps to commute to office. In the evening, she walks `40` steps per minute during the free time available every day. How many steps she walks in a day in which free time was `t` minutes?

- `4t`
- `40 t + 5000`
- `40 t + 5000`

The answer is '`40 t + 5000`'

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