Informative line

# 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.

#### 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.

#### 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

• 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.

$$x+ \not{8}- \not{8} = 12-8$$

$$\Rightarrow x=4$$

#### 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.

$$z- \not{9}+ \not{9} = 25+9$$

$$\Rightarrow z=34$$

#### 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 $$y.$$

(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.

#### 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 $$y.$$

(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.

$$\dfrac{ \not{2}b}{ \not{2}} = \dfrac{ \not{4}^2}{ \not{2}}$$

$$\Rightarrow b=2$$

#### 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.

$$\dfrac{2}{y}=\nearrow1$$

$$\Rightarrow\; 2= 1 ×y$$

$$\Rightarrow\; 2 =y$$

#### 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

#### 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