A linear equation is defined as the equation in which the highest power of each variable is one that can never be more than that.

We have already learnt about simple linear equations and the basic concepts in the previous lesson.

Today in this lesson we are going to be looking at linear equations in one variable, how can we solve linear equations in one variable and what are the methods to solve linear equations in one variable.

## Linear Equations in One Variable Definition

A linear equation in one variable is an equation which can be written as ax + b = 0 or ax = c. Where X is the variable, a is not equal to zero a,b,c are the constants (real numbers).

The equation involving only one variable in first order is called a linear equation in one variable.

Examples,
• 3x - 5 = 0
• 8 - y = 2
• 7a + a = 15 - 3a

If you notice the examples above, there are only one variable and linear in nature. This type of equation has only one solution.

## Standard Form of Linear Equations in One Variable

We have already seen above the standard form of linear equations in one variable is represented by,

ac + b = 0   or   ax = c

Where,
• a,b,c are the constants (real numbers)
• a is not equal to zero and,
• x is the variable

## Properties of an equation

1. If the same quantity is added to both sides of the equation, the sums are equal on both the sides.

Example: x = 5
xa = 5a (a added to both the side)

2. If the same quantity is subtracted from both the sides of an equation, the differences are equal.

Example: x = 5
x - a = 5 - a (a subtracted from both the sides)

3. If both the sides of an equation are multiplied by the same quantity, the multiplication of both the sides are equal.

Example: x = 5
xa = 5a (a multiplied by both the sides)

4. If both the sides of an equation are divided by the same quantity, the quotients are equal.

Example: x = a
x÷a = 5÷a (a is divided by both the sides)

These are the four properties of equations and with the help of the properties you can solve linear equations. There is a short-cut method or transposing method which is widely used to solve linear equations in one variable.

## Solving Linear Equations in One Variable

To solve a linear equation in one variable there are some steps you can follow. The steps will also help you to solve the word problems of linear equations in one variable.
1. Simplify both the sides of the equation
2. Use the addition and subtraction properties to get all the variable terms on the left hand side (LHS) and all constant terms on the right hand side (RHS).
3. Again simplify both the sides of the equation
4. Divide both sides of the equation by the coefficient of the variable.

Let's try to understand these steps with the proper example problem.

Example: Solve x + 1 = 3(x - 5).

Step 1: Simplify right hand side.
x + 1 = 3x - 15

Step 2: Substrates 3x from both the sides
x = 3x - 16

Step 3: Subtract 3x from both the sides
-2x = -16

Step 4: Divide both the sides by -2
x = 8

The solution is 8. Hence, the linear equation in one variable x + 1 = 3(x + 5) is true.

## Shortcut Method or Transposing Method

In shortcut method, there are mainly four conditions we are going to see below,

Step 1: In an equation, an added term is subtracted when it is transposed (taken) to the other side.

Example: Solve x + 2 = 10
x = 10 - 2
x = 8

Here, 2 is transposed to the other side and it is subtracted from 10.

Step 2: In an equation, a subtracted term is transposed to the other side, it is added.

Example: x - 2 = 10
x = 10 + 2 (2 is transposed)
x = 12

Step 3: In an equation, when a term in multiplication is transposed, it is divided.

Example: Solve 6x = 12
x = 12/6 (6 is transposed)
x = 2

Step 4: In an equation, when a term in division is transposed to the other side, it is multiplied.

Example: Solve x/4 = 3
x = 3×4 (4 is transposed)
x = 12

## Solved Problems of Linear Equations in One Variable

Now try to solve some problems of linear equations in one variable. Here, we will follow the shortcut method or transposing method to solve the problems.

1. Solve: 3x + 5 = 8
3x = 8 - 5 (5 is transposed)
3x = 3
x = 3/3 (3 is transposed)
x = 1

2. Solve: 2y - 4 = 6
2y = 6 + 4 (4 is transposed)
2y = 10
y = 10/2 (2 is transposed)
y = 5

3. Solve: 8x - 3 = 4x + 5
Now, transpose the variables on same side
8x - 4x = 5 + 3 (4x and 3 is transposed)
4x = 8
x = 8/4 (4 is transposed)
x = 2

### Linear Equations in One Variable Word Problems

1. A number increased by 5 is 12. Find the number?

Solution: let's the number be 'x'

Now make the equation whatever the question describes,

x + 5 = 12
x = 12 - 5
x = 7

The number is 7.

2. A number decreased by 2 is 7. Find the number?

Solution: let's the number be 'y'

y - 2 = 7
y = 7 + 2
y = 9

The number is 9.