Informative line

# Variables (Literals)

• Variables are the alphabets or letters used to represent unknown numbers/unknown quantities.
• These are also known as literals.
• Generally, we use $$x$$ or $$y$$ to represent unknown quantities, but any letter can be used as a variable.

For example: $$a,\;m,\;z,$$ etc can be used as variables.

• Variables are mostly written in lower case.
• Sometimes, figures like $$\Box,\;\Delta,$$ etc are used in place of variables.

#### Choose the variable in $$5x+10$$

A $$5$$

B $$10$$

C $$x$$

D $$+$$

×

Given: $$5x+10$$

Here, $$x$$ is used to represent an unknown number.

Thus, $$x$$ is a variable.

Hence, option (C) is correct.

### Choose the variable in $$5x+10$$

A

$$5$$

.

B

$$10$$

C

$$x$$

D

$$+$$

Option C is Correct

# Expression

• A variable can be used in any sort of mathematical expression.

## Mathematical expression

• An expression having numbers with one or more operations and no equals sign is called mathematical expression.

Example: $$2(8+6)$$

### Variable expression/Algebraic expression

• An expression having variables with one or more operations along with numbers and no equal sign is called variable expression/algebraic expression.

Example: $$6x+7,\;2z,\;3y-3$$ etc.

#### Which one of the following is an algebraic expression?

A $$3(4+7)$$

B $$71(21\div7)$$

C $$2x+1$$

D $$27(21-17)$$

×

An algebraic expression contains variables, with one or more operations along with numbers and does not contain an equals sign.

Options (A), (B) and (D) have numbers but no variables, so all three are mathematical expressions.

Hence, options (A), (B) and (D) are incorrect.

Option (C) has variable $$x$$ and numbers are connected with operations, so it is an algebraic expression.

Hence, option (C) is correct.

### Which one of the following is an algebraic expression?

A

$$3(4+7)$$

.

B

$$71(21\div7)$$

C

$$2x+1$$

D

$$27(21-17)$$

Option C is Correct

# Addition of Literals and Numbers

• Different operations can be performed on variables.
• To write an expression for a given statement, follow three steps:

(i) Identify the given number.

(ii) Identify any operations involved.

(iii) Identify the variable.

• A variable can be added to numbers.
• The addition is shown by a '$$+$$' sign in between a variable and a number.
• Words like, 'added to', 'sum', 'more than', 'increased', 'altogether', indicate addition.

For example:

1. "A variable is added to $$5$$"

Thus, it can be written as $$x+5$$.

2. "The sum of a variable and $$7$$ "

Here, sum indicates addition.

Thus, it can be written as $$x+7$$.

3. "$$y$$ more than $$x$$"

Here, more than indicates addition.

Thus, it can be written as $$x+y$$.

Note: - By commutative property $$x+y$$ is same as $$y+x$$.

#### Which expression represents the sum of a literal and $$10$$?

A $$10x$$

B $$10+x$$

C $$10-x$$

D $$\dfrac{10}{x}$$

×

Given:

"The sum of a literal and $$10$$"

Here, $$10$$ is a known number and sum indicates the addition.

A quantity which is unknown, let it be $$x.$$

Thus, it can be written by using a ' $$+$$' sign.

$$=10+x$$

Hence, option (B) is correct.

### Which expression represents the sum of a literal and $$10$$?

A

$$10x$$

.

B

$$10+x$$

C

$$10-x$$

D

$$\dfrac{10}{x}$$

Option B is Correct

# Subtraction of Literals and Numbers

• Different operations can be performed on variables.
• To write an expression for a given statement, follow three steps:

(i) Identify the given number.

(ii) Identify any operations involved.

(iii) Identify the variable.

• The variables can be subtracted from numbers and vice-versa.
• The subtraction can be shown by using a '–' sign between the variables and numbers.
• Words like, 'less than', 'taken away', 'subtracted', 'decreased', indicate subtraction.

For example:

(1) "$$5$$ less than a variable $$x$$

Here, less than indicates subtraction.

Thus, it can be written as $$x-5$$.

(2) "$$y$$ is taken away/fewer than $$x$$"

Here, taken away/fewer than indicate subtraction.

Thus, it can be written as $$x-y$$.

(3) "$$2$$ subtracted from $$y$$"

Here, subtracted indicates subtraction.

Thus, it can be written as $$y-2$$.

Note: Always take care that $$5$$ subtracted from $$x$$ represents  $$(x-5)$$. It is not same as $$x$$ subtracted from 5 which is $$(5-x)$$

#### Which expression represents $$10$$ less than a literal?

A $$10-x$$

B $$x-10$$

C $$x+10$$

D $$10\,x$$

×

Given:

"$$10$$ less than a literal"

Here,

(i) $$10$$ is a known number.

(ii) Less than indicates subtraction.

(iii) A number which is unknown, let it be $$x.$$

Thus, it can be written by using a '–' sign.

$$=x-10$$

Hence, option (B) is correct.

### Which expression represents $$10$$ less than a literal?

A

$$10-x$$

.

B

$$x-10$$

C

$$x+10$$

D

$$10\,x$$

Option B is Correct

# Multiplication of Literals with Numbers

• Different operations can be performed on variables.
•  To write an expression for a given statement, follow three steps:

(i) Identify the given number.

(ii) Identify any operations involved.

(iii) Identify the variable.

• A variable can be multiplied by a number.
• The multiplication is shown by either '$$.$$' or ' $$×$$ ' sign between a literal and a number.
• In algebra, we don't use multiplication symbol $$(×)$$ between numbers and letters because that can be very confusing with the variable $$x$$.
• Words like, 'product', 'times', indicate multiplication.

For example:

1. "$$4$$ times $$x$$"

Here, times indicates multiplication.

Thus, it can be written as $$4x$$.

2. "The product of $$x$$ and $$y$$"

Here, product indicates multiplication.

Thus, it can be written as $$xy$$.

Note: By commutative rule, $$xy$$ is same as $$yx$$.

#### Which expression represents $$1.7$$ times of a literal?

A $$1.7x$$

B $$1.7+x$$

C $$1.7-x$$

D $$\dfrac{1.7}{x}$$

×

Given:

"$$1.7$$ times of a literal"

Here,

(i) $$1.7$$ is a known number.

(ii) times indicates multiplication.

(iii) A number which is unknown, let it be $$x$$.

Thus, it is represented as $$1.7x$$.

Hence, option (A) is correct.

### Which expression represents $$1.7$$ times of a literal?

A

$$1.7x$$

.

B

$$1.7+x$$

C

$$1.7-x$$

D

$$\dfrac{1.7}{x}$$

Option A is Correct

# Division involving Literals and Numbers

• Different operations can be performed on variables.
• To write an expression for a given statement, follow three steps:

(i) Identify the given number.

(ii) Identify any operations involved.

(iii) Identify the variable.

• A variable can be divided by a number and vice-versa.
• The division is shown either by a '$$\div$$' sign or by a fraction bar $$(-)$$.
• Words like, 'by', 'divided', indicate division.

For example:

1. "$$x$$ divided by $$y$$"

Here, divided indicates division.

Thus, it can be written as $$\dfrac{x}{y}$$.

2. "$$x$$ by $$7$$"

Here, by indicates division.

Thus, it can be written as $$\dfrac{x}{7}$$.

3. "$$30$$ divided by $$x$$"

Here, divided indicates division.

Thus, it can be written as $$\dfrac{30}{x}$$.

Note: It should be taken care of that 10 by $$x$$ represents $$(\dfrac{10}{x})$$ which is not same as $$x$$ by 10 i.e., $$(\dfrac{x}{10}).$$

#### Which expression represents $$2$$ divided by a literal?

A $$2x$$

B $$2+x$$

C $$\dfrac{2}{x}$$

D $$2-x$$

×

Given:

"$$2$$ divided by a literal"

Here,

(i) $$2$$ is a known number.

(ii) Divided indicates division.

(iii) A number which is unknown, let it be $$x.$$

Thus, it can be written as $$\dfrac{2}{x}$$.

Hence, option (C) is correct.

### Which expression represents $$2$$ divided by a literal?

A

$$2x$$

.

B

$$2+x$$

C

$$\dfrac{2}{x}$$

D

$$2-x$$

Option C is Correct