LINEAR EQUATION IN ONE VARIABLE WITH EXAMPLES AND FAQ

LINEAR EQUATION IN ONE VARIABLE

Linear equation in one variable is an equation that is usually expressed in the form:

ay + b = 0

Where a and b are real numbers and y is a variable. This form of a linear equation has only one solution.

When the linear equation is plotted on a graph, the graph represents a straight line. The straight line plotted will be parallel to any of the axes. While plotting the linear equation on the x-y plane, if the variable in the equation is ‘y’, then the straight line will be parallel to the x-axis. Similarly, if the variable in the equation is ‘x’, then the straight line will be parallel to the y-axis. The solution of the equation will be the point where the straight line intersects any of the two axes.

For example, 14y + 8 = 20 is a linear equation having a single variable ‘y’ in it.

Here the variable is ‘y’ hence, the straight line is parallel to the x-axis. The solution of this line is the point where the straight line intersects the y-axis.

What is a linear equation?

A linear equation is a type of equation in which the degree of each variable in the equation is equal to one.

For example, 3x + 20 = 5 is a linear equation as the degree of ‘x’ is 1. Whereas, an equation, x^{2}+16=20 is not a linear equation as the degree of ‘x’ is 2 in this case. 

Standard Form 

The general form is ax + c = 0. Here a is the coefficient of x.  x is the variable. c is the constant term. The coefficient and the constant term should be separated to find the final solution of this linear equation.

Solving linear equation in one variable involves the following steps:

• The variables should be brought to one side of the equation.
• The constant terms are moved to the other side of the equation.
• If the variable has a coefficient, then the equation is divided on both sides by the coefficient.

For the general form, i.e., ax + c = 0,

First, the constant term ‘c’ is moved to the right side,

\Rightarrow a x=-c

Now to find the value of x we divide the above equation on both sides by ‘a’,

\Rightarrow \mathrm{x}=-\frac{c}{a}

which is the solution of the equation.

Examples

Example 1: Consider the equation: 14x – 18 = 8x + 42

Step 1: Isolate all the variables on one side of the equation i.e., shift the variables from one side of the equation to the other side of the equation.

In the equation 14x – 18 = 8x + 42, we transpose 8x from the right-hand side to the left-hand side of the equality, the operation gets reversed upon transposition and the equation becomes:

 14x – 18 – 8x = 42

\Rightarrow 6 x-18=42

Step 2: Similarly shifting the constant terms to the other side of the equation:

   6x – 18 = 42

 ⇒ 6x = 42 + 18

 ⇒ 6x = 60

Step 3: Divide the equation with 6 on both sides of the equality.

\Rightarrow \frac{6x}{6}=\frac{60}{6}

\Rightarrow x=10

If we substitute x = 10 in the equation 14x –18 = 8x + 42, we will get 122 = 122  which satisfies the equality and hence is the required solution.

Example 2: Fifteen years ago, Arjun was one-fourth of what his age is now. How old is Arjun?

Solution:

Let Arjun’s present age be x years.

Hence, fifteen years ago, Arjun’s age was (x-15) years. According to the given information,

x – 15 = x/4

⇒ 4(x – 15) = 4x/4           [Multiplying 4 on both sides of the equation]

⇒ 4x – 60 = x                   [Simplifying the equation]

⇒ 4x – x = 60

⇒ 3x = 60

⇒ x = 60/3

Hence, x = 20.

Therefore, Arjun is 20 years old.

Example 3: Find two numbers such that one of the numbers is 6 more than the other and the sum of the two numbers is 28.

Solution: 

We do not know either of the two numbers, and we need to find them.

Two conditions are given:

1. One of the numbers is 6 more than the other.
2. The sum of the two numbers is given to be 28.

Let the smaller number be x, then the larger number is 6 more than x, i.e., x  + 6.

The other condition says that the sum of these two numbers x and x  + 6 is 28.

This means that  x  + (x  + 6) = 28

2 x+6=28 \Rightarrow 2 x=28-6 = 22

\therefore x=\frac{22}{2}=11

Now, x + 6 = 11 + 6 = 17

Therefore, the two numbers which have a difference of 6 and their sum is 28 are 11 and 17.