**Linear Equation in One Variable**

** 1. Definition of an Equation:**

What is an equation?

A statement of equality of two algebraic expressions in or more variables is called an equation.

*Examples: *
1. x + 1 = 2 and

2. 2y + 3 = 5

In 1 above, x + 1 is an algebraic expression in the variable x. we read it as “ x plus 1 is equal to 2”.

x is some number, called variable.

In 2 above, 2y + 3 is an algebraic expression in the variable y, we read it as “ 2 times y or 2y plus 3 is equal to 5”. y is some number, called variable

**2. The three components of an equation**

Every equation has a Left hand side, the equality sign ** ‘=’** and the Right hand side.
*
the three components in the equation x + 1 = 2 are :*

**
L.H.S = Left Hand Side
R.H.S = Right Hand Side**

**3. Solution/Root of an Equation: **

the value of x, i.e. some number for x, which makes the equation a true statement is called solution or *root of the equation. *

In simple words, if the L.H.S. and R.H.S become equal for some number plugged in for x, then the number, also called value, is the solution or *root of the equation. *

In x + 1 = 2, what should be plugged in for x so that L.H.S. becomes equal to R.H.S?

It is 1, i.e. if 1 is plugged in for x, the two sides become equal.

*This number or value 1 for x is called root or solution of the equation. Transposition Rule:*

**
Transposition Rule**

1. The transposition rule applies on addition and subtraction.

Terms can be transposed (shifted) between either sides of the equality symbol “=” with a change in sign of the transposed terms”

*Example:*
x + 1 = 2.

1 can be transposed to right side by inversing its sign

i.e. x = -1 + 2
so that we have x = 1

transposition conforms with the rule:

* “same numbers can be added on both sides of an equation”*

So, in x + 1 = 2, to find x, we need to get rid of 1 on the left hand side, to do this add -1 on both sides of the equation.

x +1-1 = -1 + 2,

x = 1

**
Application of Linear Equations or Word Problems on Linear Equations in one Variable **

**Problem 1:
The sum of two consecutive numbers is 25. Find the numbers. **

*Solution:*

Let the two consecutive numbers be x andx+1.

So we can set up the following linear equation:

Given that x + x+1= 25,

2x = 24, {from transposition rule, inverse sign of 1 on taking it to right side}

x = 24/2 = 12, {multiplication changes to division}

So the other number is x + 1 = 12 + 1 = 13

Therefore, the two numbers are 12 and 13

Problem 2:

John is 15 years now. He is 10 years older than his brother Tom. How old is Tom 10 years from now?

*Solution:*
Let Tom be x years old now.

So, we set up the linear equation:

x + 10 = 15, x = 5. {from transposition rule, inverse sign of 10 on taking it to right side}

So, tom is 5 years old now.

10 years from now, tom will be 10 + 5 = 15

Problem 3:

20 years from now, Nancy will become three times as old as he is now. Find her age now.

*Solution: *

Let Nancy be x years old now.

20 years from now, she will be x +20

But 20 years from now, she we thrice her present age, x i.e. 3x

So, x + 20 = 3x, or 3x = x + 20,

3x – x = 20, {from transposition rule}

2x = 20,

x = 20/2 = 10