- An equation is like a weighing scale. Both sides should always be perfectly balanced.
- Using equations, we can express math facts. In short, solve problems easily and quickly.
- It is a mathematical expression that contains an equals "=" sign.

- There are two types of equations.

**(1) Numerical equation- **A mathematical statement that contains numbers and equal "=" sign.

**For example:** \(4+6=10\)

**(2) Algebraic equation-** A mathematical statement that contains numbers, variables and equal "=" sign.

**For example:** \(2x+3=13\)

A \(2+3=1+5\)

B \(3x+6\)

C \(36x+2=10\)

D \(5+6\)

- An equation can be written in an algebraic form from verbal models (phrases) with the help of keywords.
- Following keywords help us to write the statements.

**1. Keywords for Addition:**

Plus

More than

And

Sum

Total

Added to

Combine

Altogether

**2. Keywords for Subtraction:**

Minus

Less than

Fewer than

Difference

Reduce

**3. Keywords for Multiplication:**

Times

Product

Double (multiplied by \(2\))

Twice (multiplied by \(2\))

Triple (multiplied by \(3\))

Quadruple (multiplied by \(4\))

**4. Keywords for Division:**

By

Divided into

Shared

Quotient

Per

Share (equally)

Split (Equally)

Half (divided by \(2\))

**For ****example :** Thirteen times a number is twenty-six. Write it as a mathematical equation.

Here,

Thirteen \(\to\;13\)

Times \(\to\;×\) (multiplication)

a number \(\to\;let\;(x)\)

is \(\to\;=\)

Twenty-six \(\to\;26\)

- So, our equation is-

\(13×x=26\;\text{or}\;13x=26\)

**Examples:**

- Two less than a number is five, means \(x-2=5\)
- Half of a number is nine, means \(x\div2=9\)
- Ten and a number is fifteen, means \(10+x=15\)

A \(22\div x=28\)

B \(22x=28\)

C \(22-x=28\)

D \(22+x=28\)

- We can use an equation to describe a real world situation.

**For example:**

On his birthday, Aron brought \(100\) candies to school. After distribution, he was left with \(20\) candies. Write an equation to represent the number of candies he distributed.

\(\to\) Let number of candies Aron distributed \(=x\)

\(\to\) Number of candies Aron brought \(=100\)

\(\to\) Number of candies he was left with \(=20\)

Here,

\(\underbrace{\text{(Number of candies Aron brought)}}_{100}-\underbrace{\text{(Number of candies he distributed)}}_{x}=\underbrace{\text{(Number of candies he was left with)}}_{20}\)

So, the equation is, \(100-x=20\)

A \(2x-6=8\)

B \(8-6x=2\)

C \(2x-8=6\)

D \(8x-6=2\)

- Equations can be written in statements (phrases) with the help of keywords:
- Following keywords help us to write the statements.

**1. Keywords for Addition:**

Plus

More than

And

Sum

Total

Added to

Combine

Altogether

**2. Keywords for Subtraction:**

Minus

Less than

Fewer than

Difference

Reduce

**3. Keywords for Multiplication:**

Times

Product

Double (multiplied by \(2\))

Twice (multiplied by \(2\))

Triple (multiplied by \(3\))

Quadruple (multiplied by \(4\))

**4. Keywords for Division:**

By

Divided into

Shared

Quotient

Per

Share (equally)

Split (Equally)

Half (divided by \(2\))

**For example :**

Write \(6a=b\) in words.

\(6a=b\;\text{or}\;6×a=b\)

Here,

\('6'\;\to\) Six

Multiplication \((\times)\;\to\) times

\('a'\;\to\) a number

\('='\;\to\) is

\('b'\;\to\) another number

So, the equation is-

Six times a number is another number.

**Examples:**

- \(\dfrac{x}{2}-7=y\) means Seven less than half of a number is another number.
- \(a+b=10\) means Sum of two numbers is ten.
- \(2x+2=y\) means Twice a number and two is another number.

A The sum of a number and the quotient of another number by five is twelve

B Five divided by a number plus another number is twelve

C A number and five plus another number is twelve

D five times a number plus another number is twelve

- Forming an equation from real world situation can be understood by the following example.

There are \('b'\) number of packets of chocolates in a box. Each packet has \('x'\) chocolates and it is given that a total number of chocolates in the box are \('y'\) . Write an equation which represents the given situation.

\(\to\) In the given situation, three variables are used which are as follows-

\('y'\) represents the total number of chocolates in the box.

\('b'\) represents the total number of packets in the box.

\('x'\) represents the total number of chocolates in each packet.

\(\to\) Since there are \('b'\) number of packets and each has \('x'\) number of chocolates. Thus, the total number of chocolates will be equal to the product of number of packets and number of chocolates in each packet.

\(\therefore\) Total number of chocolates in the box = (Total number of packets in the box) × (Total number of chocolates in each packet)

\(\Rightarrow\;y=b×x\)

or

\(\Rightarrow\;y=bx\)

A \(x=70+y\)

B \(70=x+y\)

C \(y=70+x\)

D \(x+70=70+y\)

- We can use an equation to describe a real world situation.
- We can make a real world situation through the equation too.
- Consider an example to understand it easily.

\($x=$(2y-50)\)

Here, we will make a real world situation for the given equation.

Cody has \($x\) that is equal to \($50\) less from the two times of money \($y\).

A Cost of a pen is \($x\), then cost of \(y\) pens is \($35\).

B Cost of a black pen and a blue pen is \($x\) and \($y\) respectively. The total cost of both pens is \($35\).

C Cooper had \($x\), after purchasing a pen for \($y\) he was left with \($35\).

D Mr. Gibbons distributed \(x\) pens to \(y\) students so that each student had \(35\) pens.

- Equations can be written in statements (phrases) with the help of keywords:
- Following keywords help us to write the statements.

**1. Keywords for Addition:**

Plus

More than

And

Sum

Total

Added to

Combine

Altogether

**2. Keywords for Subtraction:**

Minus

Less than

Fewer than

Difference

Reduce

**3. Keywords for Multiplication:**

Times

Product

Double (multiplied by \(2\))

Twice (multiplied by \(2\))

Triple (multiplied by \(3\))

Quadruple (multiplied by \(4\))

**4. Keywords for Division:**

By

Divided into

Shared

Quotient

Per

Share (equally)

Split (Equally)

Half (divided by \(2\))

**For example:** write the equation, \(9-x=7\) in words.

Here,

\(9\;\to\) Nine

\(-\;\to\) less than

\(x\;\to\) a number

\(=\;\to\) is

\(7\;\to\) Seven

- So, the equation can be stated as-

A number less than nine is seven.

**Example:**

- \(6+x=10\) means six and a number is ten
- \(4x=12\) means quadruple of a number is twelve
- \(x\div2=6\) means half of a number is six

A Five shared equally by a number is eight hundredths

B Five minus a number is eight hundredths

C Five times a number is eight hundredths

D Five and a number is eight hundredths

- An equation can be written in an algebraic form from verbal models (phrases) with the help of keywords.
- Following keywords help us to write the statements.

**1. Keywords for Addition:**

Plus

More than

And

Sum

Total

Added to

Combine

Altogether

**2. Keywords for Subtraction:**

Minus

Less than

Fewer than

Difference

Reduce

**3. Keywords for Multiplication:**

Times

Product

Double (multiplied by \(2\))

Twice (multiplied by \(2\))

Triple (multiplied by \(3\))

Quadruple (multiplied by \(4\))

**4. Keywords for Division:**

By

Divided into

Shared

Quotient

Per

Share (equally)

Split (Equally)

Half (divided by \(2\))

**For example:**

'Seven times a number is another number'. Write it as a mathematical equation.

Here,

Seven \(\to\;7\)

Times \(\to\;\times\) (multiplication)

a number \(\to\;let\;(x)\)

is \(\to\;=\)

another number \(\to\;let\;(y)\)

So, the equation is-

\(7×x=y\;\text{or}\;7x=y\)

**Examples:**

- Twice of a number is six times another number, means \(2x=6y\)
- Nine less than a number is another number, means \(x-9=y\)
- A number divided by another number is two, means \(x\div y=2\)

A \(x-2=y\)

B \(x+2=y\)

C \(x×2=y\)

D \(x\div2=y\)