Informative line

Equation And Its Solution (inverse Operation)

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.
  • Definition: An equation is a mathematical expression that contains an equals sign, "=" .
  • An equation says that two things are equal.

For example:

\(x+3=12\)

  • Here, one side \((x+3)\) is equal to another side \((12).\)
  • In an equation, a variable shows an unknown quantity which can be calculated by solving the equation.
  • In an equation, an equals sign is used for balancing two expressions.

Illustration Questions

Which one of the following is an equation?

A \(x>12\)

B \(3x<4\)

C \(3x=12\)

D \(4x+6\)

×

An equation always contains an equals sign, "=" .

Options (A) and (B) show inequalities because they have inequality symbols of greater than  ">" and less than "<" respectively.

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

Option (C) has an equals sign, which shows that \(3x\) is equal to \(12\).

Hence, option (C) is correct.

Option (D) represents an expression, because an expression does not contain an equals sign.

Hence, option (D) is incorrect.

Which one of the following is an equation?

A

\(x>12\)

.

B

\(3x<4\)

C

\(3x=12\)

D

\(4x+6\)

Option C is Correct

Writing an Equation in One Variable

  • Using equations we can solve many mathematical and daily life problems.
  • An equation should have variables, constants and an equals sign.
  • Variables are used for unknown quantities.
  • If we have one unknown quantity, we use one variable and if we have more than one unknown quantities, we use variables accordingly.
  • Writing an algebraic equation involves translating a written statement into an algebraic form.
  • To write an equation from the given data, go step by step according to the statement.
  • Some important phrases- 
  1. For addition - Sum, plus, altogether, and, more than
  2. For subtraction - Difference, less than, subtract, take away
  3. For multiplication - Product, times
  4. For division - Split up, quotient, divided, share
  • To make an equation, identify the following elements:
  1. Identify the numbers.
  2. Identify the operation involved.
  3. Identify the variable.

Consider an example:

(i) The product of five and a number is \(15\).

  • Here, product means multiply.
  • A number which is unknown, assume it to be \(x\).
  • ''Is'' means equals.
  • \(5\) is a number.

Now go step by step, multiply \(5\) and \(x.\)

That means \(5x\)

The result is \(15\).

So, we can write

\(5x=15\)

This is the required equation.

Illustration Questions

If a number is divided by \(4,\) we get \(20.\) Which equation correctly represents the given statement?

A \(5+x=20\)

B \(4x=20\)

C \(4-x=20\)

D \(\dfrac{x}{4}=20\)

×

Given: If a number is divided by \(4,\) we get \(20.\)

 

An equation should have variables, constants and an equals sign.

(i) Here, divided means division of a number by \(4.\)

(ii) A number which is unknown, assume it to be \(x.\)

(iii) \(4\) is a number.

(iv) Result is \(20\), represent it by an equals sign.

Now going step by step, dividing \(x\) by \(4.\)

That means \(\dfrac{x}{4}\)

The result is \(20\).

\(\dfrac{x}{4}=20\)

This is the required equation.

Hence, option (D) is correct.

If a number is divided by \(4,\) we get \(20.\) Which equation correctly represents the given statement?

A

\(5+x=20\)

.

B

\(4x=20\)

C

\(4-x=20\)

D

\(\dfrac{x}{4}=20\)

Option D is Correct

Solving Equations Involving Addition

  • Till now we have learnt how to write an equation in one and two variables.
  • Now, we will take one step ahead and learn solving equations in one variable.
  • Solving an equation means we will find out the value of the variable. 
  • Here, we will learn to solve equations involving only one operation i.e. addition.
  • We will use inverse operations to solve equations.
  • Inverse operation: An inverse operation is the opposite of the given operation.

For example: Solve for \(x\);

\(x+8 = 12\)

We know that the inverse operation of addition is subtraction.

Thus, we will subtract \(8\) from both sides of the equation to get only the variable on one side.

\(x+ \not{8}- \not{8} = 12-8\)

\(\Rightarrow x=4\)

Illustration Questions

Solve for \(x\):  \(x+15=9\)

A \(10\)

B \(6\)

C \(15\)

D \(-6\)

×

Given equation: \(x+15=9\)

The inverse operation of addition is subtraction.

We will subtract \(15\) from both sides of the equation to get only the variable on one side.

\(x+15-15=9-15\)

\(\Rightarrow x= -6\)

Hence, option (D) is correct.

Solve for \(x\):  \(x+15=9\)

A

\(10\)

.

B

\(6\)

C

\(15\)

D

\(-6\)

Option D is Correct

Solving Equations Involving Subtraction

  • Here, we will learn to solve equations involving only one operation i.e. subtraction. 
  • We will use inverse operations to solve equations. 
  • Inverse operation: An inverse operation is the opposite of the given operation.

For example: Solve for \(z; \)

\(z-9=25\)

We know that the inverse operation of subtraction is addition.

Thus, we will add \(9\)to both sides of the equation to get only the variable on one side.

\(z- \not{9}+ \not{9} = 25+9\)

\(\Rightarrow z=34\)

Illustration Questions

Solve for ​​\(a\): \( a-6 = 1\)

A \(6\)

B \(7\)

C \(5\)

D \(1\)

×

Given equation: \(a-6=1\)

The inverse operation of subtraction is addition.

Thus, we will add \(6\) to both sides of the equation to get only the variable on one side.

\(a-6=1\)

\(a- \not{6}+ \not{6}=1+6\)

\(\Rightarrow a=7\)

Hence, option (B) is correct.

Solve for ​​\(a\): \( a-6 = 1\)

A

\(6\)

.

B

\(7\)

C

\(5\)

D

\(1\)

Option B is Correct

Writing an Equation in Two Variables

  • Writing an algebraic equation involves translating a written statement into an algebraic form.
  • If we have two unknown quantities, we use two variables to write an equation from the given data.

Look at an example:

Olive has some pencils and some pens, altogether \(20.\) To write it in equation form, identify the important elements:

(i) Here, the number of pens and pencils are two unknown quantities, so we use two variables, \(x\) and \(y.\)

(ii) Altogether means addition.

(iii) "Is" means equals.

Now go step by step, add \(x\) and \(y.\)

That means \(x+y\)

The result is \(20\).

\(\therefore\) We can write,

\(x+y=20\)

This is the required equation.

Illustration Questions

Charlie has some red balls and white balls, altogether \(10.\) Which equation correctly represents the given statement?

A \(x-y=20\)

B \(x+y=10\)

C \(\dfrac{x}{y}=10\)

D \(xy=10\)

×

An equation should have variables, constants and an equals sign.

(i) Here, the number of red balls and white balls are two unknown quantities, so we use two variables, \(x\) and \(y.\)

(ii) Here, altogether means addition.

(iii) The result is \(10\), represent it by an equals sign.

Now going step by step, addition of \(x\) and \(y\)

That means, \(x+y\)

Altogether \(10\)

So, we can write

\(x+y=10\)

This is the required equation.

Hence, option (B) is correct.

Charlie has some red balls and white balls, altogether \(10.\) Which equation correctly represents the given statement?

A

\(x-y=20\)

.

B

\(x+y=10\)

C

\(\dfrac{x}{y}=10\)

D

\(xy=10\)

Option B is Correct

Solving Equations Involving Multiplication

  • Here, we will learn to solve equations involving only one operation, i.e. multiplication.
  • We will use inverse operations to solve equations. 
  • Inverse operation: An inverse operation is the opposite of the given operation.

For example: Solve for \(b;\;2b=4\)

We know that the inverse operation of multiplication is division.

Thus, we will divide by \(2\) on both sides of the equation to get only the variable on one side.

\(\dfrac{ \not{2}b}{ \not{2}} = \dfrac{ \not{4}^2}{ \not{2}}\)

\(\Rightarrow b=2\)

Illustration Questions

Solve for \(m\):  ​​​\(5m=3.25\)

A \(0.62\)

B \(3.3\)

C \(0.65\)

D \(2.0\)

×

Given equation: \(5m=3.25\)

The inverse operation of multiplication is division.

We will divide by \(5\) on both sides of the equation to get only the variable on one side.

\(\dfrac{ \not{5}m}{ \not{5}}= \dfrac{3.25}{5}\)

\(m = 0.65\)

Hence, option (C) is correct.

Solve for \(m\):  ​​​\(5m=3.25\)

A

\(0.62\)

.

B

\(3.3\)

C

\(0.65\)

D

\(2.0\)

Option C is Correct

Solving Equations Involving Division

  • Here, we will learn how to solve equations involving only one operation, i.e. division.
  • We will use inverse operations to solve equations. 
  • Inverse operation: An inverse operation is the opposite of the given operation.

For example: Solve for \(y:\; \dfrac{2}{y}=1\)

We know that the inverse operation of division is multiplication.

Thus, we will multiply the denominator \((y)\) with the value \((1)\) on the right side of the equation to get only the variable on one side.

\(\dfrac{2}{y}=\nearrow1\)

\(\Rightarrow\; 2= 1 ×y \)

\(\Rightarrow\; 2 =y\)

Illustration Questions

Solve for ​\(c\): \(\dfrac{c}{3} = 2\)

A \(4\)

B \(3\)

C \(6\)

D \(2\)

×

Given equation: \(\dfrac{c}{3} = 2\) 

The inverse operation of division is multiplication.

Thus, we will multiply \(3\) by  \(2\) which is on the right side of the equation to get only the variable on one side.

\(c= 2×3\)

\(\Rightarrow c = 6\)

Hence, option (C) is correct.

Solve for ​\(c\): \(\dfrac{c}{3} = 2\)

A

\(4\)

.

B

\(3\)

C

\(6\)

D

\(2\)

Option C is Correct

Illustration Questions

Carl has five more than twice as many peanuts as walnuts. He has \(32\) nuts in total. Which equation correctly represents the given problem?

A \(2x=32\)

B \(3x+5 = 32\)

C \(2x+5=32\)

D \(5x+2 = 32\)

×

An equation should have variables, constants and an equals sign.

Here, two operations are used. 

(i) Five more than, i.e. addition of \(5\).

(ii) Twice as many, i.e. multiplication by \(2\).

Let the number of walnuts be \(x\).

Now, the number of peanuts is \(5\) more than twice as walnuts, i.e.

\(5+2x\)

The sum of peanuts and walnuts is \(32\).

Thus, the equation becomes

\(x + 5 + 2x= 32\)

\(\Rightarrow 3x+5 = 32\)

Hence, option (B) is correct.

Carl has five more than twice as many peanuts as walnuts. He has \(32\) nuts in total. Which equation correctly represents the given problem?

A

\(2x=32\)

.

B

\(3x+5 = 32\)

C

\(2x+5=32\)

D

\(5x+2 = 32\)

Option B is Correct

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