A linear equation in one variable is an equation that has only one variable and the highest power (degree) of the variable is 1. It is used to represent situations where the value of one quantity depends on another.
Exmaples:
- 2a + 3a = 20
- 5x + x = 12, etc.
The above equations are linear in one variable as they only have one variable and the highest degree of the variable is 1.
Graph of Linear Equation in One Variable
These linear equation can be easily represented on the graphs and they represent the straight line which can be horizontal to the coordinate axis or vertical to the coordinate axis.
We represent the mathematical condition using these equations as the condition "a number which is 5 less than twice of itself" is represented by the linear equation,
x = 2x - 5 , Where x is the unknown number.
In the above equation there is only one variable (x) and the degree of the variable is one thus, it is a linear equation in one variable.
Standard Form of Linear Equation in One Variable
A linear equation in one variable can be expressed in the standard form
ax + b = 0
where x is the variable and a and b are the constants involved. These constants (a and b) are non-zero real numbers. These equations have only one possible solution for the value of the variable.
Solving Linear Equations in One Variable
Linear Equations in One Variable can easily be solved by following the steps discussed below,
- Step 1: Write the given equation in standard form and if a or b is a fraction then take the LCM of the fraction to make them integers.
- Step 2: The constants are then taken to the right side of the equation.
- Step 3: All the variables and the constant terms are then simplified to form one single variable term and one single constant.
- Step 4: The coefficient of the variable is made 1 by dividing both sides with a suitable constant to get the final result.
Example: Solve 3x + 1/2 = (1/2)x - 13/2
Step 1: Arranging in standard form and taking LCM
3x - (1/2)x + 1/2 + 13/2 = 0
⇒ (6x - x + 1 + 13)/2 = 0
⇒ 6x - x + 1 + 13 = 0
Step 2: Transposing the constant to the right-hand side
6x - x = -1 - 13
Step 3: Simplification
5x = -14
Step 4: Make coefficient of x to 1
Dividing both sides by 5 we get,
5x/5 = -14/5
x = -14/5
This is the required solution.
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Linear Equation in One Variable Examples
Example 1: Solve for y, 8y - 4 = 0
Solving for value of y,
Adding 4 to both sides of the equation ,
8y -4 + 4 = 4
8y = 4Dividing both sides of equation by 8
y = 4/8Simplifying the equation ,
y = 1/2
Example 2: Solve the equation in x, 3x +10 = 55
Taking constants to RHS,
3x = 45
x = 15
Example 3: Solve the equation in x, 4/x5 -5 = 15
4/x5 -5 = 15
⇒ 4x/5 = 15 + 5
⇒ 4x/5 = 20
⇒ x = 20×5/4
⇒ x = 25
Example 4: The age of Ravi is twice the age of his sister Kiran if the sum of their age is 24 find their individual age.
Let the age of Kiran is x, then the age of Ravi is 2x
Given, the sum of their ages is 24
x + 2x = 24
⇒ 3x = 24
⇒ x = 8Thus, the age of Kiran = x = 8 years
Age of Ravi = 2x = 2×8 = 16 years
Example 5: Akshay earns three times more than Abhay and if the difference in their salaries is Rs 5000 find their individual salaries.
Let the salary of Abhay be x, then the salary of Akshay is 3x
Given, the difference in their salaries is Rs 5000
3x - x = 5000
⇒ 2x = 5000
⇒ x = 2500Thus, the salary of Abhay = x = Rs 2500
The salary of Akshay = 3x = 3×2500 = Rs 7500
Unsolved Questions on Linear Equation in One Variable
Question 1: Five times a number reduced by 9 gives 41. Find the number.
Question 2: The sum of a number and 12 is 35. Find the number.
Question 3: The age of Ravi is twice the age of his sister Kiran. The sum of their ages is 24 years.
Find the age of Ravi and Kiran.
Question 4: The sum of the ages of a father and his son is 48 years. The father is 20 years older than the son. Find their ages.