Informative line

Formation Of Equations

Algebraic Equation

What is an equation?

  • 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.

Types of equation

  • 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: \(2x+3=13\)

Illustration Questions

Which one of the following is an algebraic equation?

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

B \(3x+6\)

C \(36x+2=10\)

D \(5+6\)

×

A mathematical statement that contains numbers, variables and equal "=" sign is called algebraic equation.

In option (A), there are numbers and equal sign, but there are no variables. Thus, it is not an algebraic equation.

Hence, option (A) is incorrect.

In option (B), there are numbers and variables, but there is no equal sign. Thus, it is not an algebraic equation.

Hence, option (B) is incorrect.

In option (C), there are numbers, variables and equal "=" sign. Thus, it is an algebraic equation.

Hence, option (C) is correct.

In option (D), there are only numbers, but there is no equal sign and no variables. Thus, it is not an algebraic equation.

Hence, option (D) is incorrect.

Which one of the following is an algebraic equation?

A

\(2+3=1+5\)

.

B

\(3x+6\)

C

\(36x+2=10\)

D

\(5+6\)

Option C is Correct

Writing Equations from Verbal Models (Phrases)

  • 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:

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

Illustration Questions

Which one of the following options represents "Twenty-two and a number is twenty-eight" as a mathematical equation?

A \(22\div x=28\)

B \(22x=28\)

C \(22-x=28\)

D \(22+x=28\)

×

Given:  Twenty-two and a number is twenty-eight.

Here,

Twenty-two \(\to\;22\)

and \(\to\;+\) (Addition)

a number \(\to\;let\;(x)\)

is \(\to\;=\) 

Twenty-eight \(\to\;28\)

So, the equation is:

\(22+x=28\)

Hence, option (D) is correct.

Which one of the following options represents "Twenty-two and a number is twenty-eight" as a mathematical equation?

A

\(22\div x=28\)

.

B

\(22x=28\)

C

\(22-x=28\)

D

\(22+x=28\)

Option D is Correct

Making Equations from Real Situations

  • 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, \(100-x=20\)

Illustration Questions

In a basketball match, Alex scored \(8\) points. He scored \(6\) fewer than twice of the number Aron did. Which one of the following equations shows, how many points did Aron score?

A \(2x-6=8\)

B \(8-6x=2\)

C \(2x-8=6\)

D \(8x-6=2\)

×

Given : Number of points Alex scored \(=8\)

Here,

Points scored by Aron \(=let\;(x)\)

Twice of the number \(=2x\)

\(6\) fewer than twice of a number \(=2x-6\)

Number of points Alex scored \(=8\)

Alex scored \(6\) fewer that twice of the number Aron did.

So, the equation is \(2x-6=8\)

Hence, option (A) is correct.

In a basketball match, Alex scored \(8\) points. He scored \(6\) fewer than twice of the number Aron did. Which one of the following equations shows, how many points did Aron score?

A

\(2x-6=8\)

.

B

\(8-6x=2\)

C

\(2x-8=6\)

D

\(8x-6=2\)

Option A is Correct

Writing Statements from Equations (two or more variable)

  • 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:

  1. \(\dfrac{x}{2}-7=y\) means Seven less than half of a number is another number.
  2. \(a+b=10\) means Sum of two numbers is ten.
  3. \(2x+2=y\) means Twice a number and two is another number.

Illustration Questions

Which one of the following options represents \('Q+\dfrac{P}{5}=12'\) in words?

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

×

Given: \(Q+\dfrac{P}{5}=12\)

Equation : \(Q+\dfrac{P}{5}=12\)

Here,

\(Q\to\) A number

\(+\to\) And

\(P\to\) another number

 \(\;/\,\to\) by

\(5\to\) five

\(=\;\to\) is

\(12\to\) twelve

So, the statement is:

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

Hence, option (A) is correct.

Which one of the following options represents \('Q+\dfrac{P}{5}=12'\) in words?

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

Option A is Correct

Forming Equations Involving two or more Variables from Real Life Problems

  • 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

\(\Rightarrow\;y=bx\)

Illustration Questions

\(70\) students are there in Mr. Larry's class. If there are \(x\) girls and \(y\) boys, the choose the correct equation to represent the given situation.

A \(x=70+y\)

B \(70=x+y\)

C \(y=70+x\)

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

×

In the given situation, following two variables are used:

\(x\) represents the total number of girls in the class.

\(y\) represents the total number of boys in the class.

Total number of students \(=70\)

Since, total students in Mr. Larry's class are equal to the sum of the number of girls and boys.

\(\Rightarrow\) Total students in Mr. Larry's class = Total number of girls + Total number of boys

Thus, the equation is

\(70=x+y\)

Hence, option (B) is correct.

\(70\) students are there in Mr. Larry's class. If there are \(x\) girls and \(y\) boys, the choose the correct equation to represent the given situation.

A

\(x=70+y\)

.

B

\(70=x+y\)

C

\(y=70+x\)

D

\(x+70=70+y\)

Option B is Correct

Expressing Equations through Real Life Problems

  • 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 \($y\).

Illustration Questions

Choose the correct situation which satisfies the following equation:   \(x+y=35\)

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.

×

For option (A)

Given: Cost of a pen is \($x\).

\(\therefore\) Cost of \(y\) pens is \(x×y\)

But according to option (A), cost of \(y\) pens is \(35\).

Thus, equation for option (A) is:

\(x×y=35\)

or

\(xy=35\)

This situation does not satisfy the given equation.

Hence, option (A) is incorrect.

For option (B)

Given: Cost of black pen is \($x\).

Cost of blue pen is \($y\).

Then total cost of both the pens is \(x+y\)

But according to option (B), total cost of both the pens is \(35\).

Thus, equation of option (B) is:

\(x+y=35\)

This situation satisfies the given equation.

Hence, option (B) is correct.

For option (C)

Given: Cooper had \($x\).

Cost of a pen is \($y\).

Thus, the amount left with Cooper after purchasing a pen is  \($(x-y)\)

But according to option (C), Cooper had \($\,35\) after he bought a pen.

Thus, equation for option (C) is:

\(x-y=35\)

This situation does not satisfy the given equation.

Hence, option (C) is incorrect.

For option (D)

Given: Mr. Gibbons distributed  \(x\) pens to \(y\) students.

It means each student had \(\dfrac{x}{y}\) pens.

But according to option (D), each student had \(35\) pens.

Thus, equation for option (D) is:

\(\dfrac{x}{y}=35\)

This situation does not satisfy the given equation.

Hence, option (D) is incorrect.

Choose the correct situation which satisfies the following equation:   \(x+y=35\)

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.

Option B is Correct

Writing Statement from Equation

  • 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:

  1. \(6+x=10\) means six and a number is ten
  2. \(4x=12\) means quadruple of a number is twelve
  3. \(x\div2=6\) means half of a number is six

Illustration Questions

Which one of the following represents \(5m=0.08\) in words?

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

×

Given: \(5m=0.08\;\text{or}\;5×m=0.08\)

Here,

\(5\;\to\) Five

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

\(m\;\to\) a number

\(0.08\to\) Eight hundredths

So, the equation represents-

Five times a number is eight hundredths.

Hence, option (C) is correct.

Which one of the following represents \(5m=0.08\) in words?

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

Option C is Correct

Writing Equations from Verbal Models (2 or more variables)

  • 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:

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

Illustration Questions

Which one of the following mathematical equations represents- "Half of a number is another number"?

A \(x-2=y\)

B \(x+2=y\)

C \(x×2=y\)

D \(x\div2=y\)

×

Given: Half of a number is another number.

Here,

a number \(\to\;let\;(x)\)

another number \(\to\;let\;(y)\)

'of' \(\to\)  multiplication \((\times)\)

Half of a number \(\to\;\dfrac{1}{2}×x=\dfrac{x}{2}=x\div2\) 

is \(\to\;=\)

So, the equation is:

\(x\div2=y\)

Hence, option (D) is correct.

Which one of the following mathematical equations represents- "Half of a number is another number"?

A

\(x-2=y\)

.

B

\(x+2=y\)

C

\(x×2=y\)

D

\(x\div2=y\)

Option D is Correct

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