What is a Linear Equation?
A linear equation is defined as an equation that has the highest degree, that is, one.
Linear equations are also known as equations of the first order or one degree equations.
For example, ax + by = c is a linear equation in which x and y are the variables, a and b are the coefficients, and c is the constant. Note that the power of the variables x and y is 1.
The graph of a linear equation is always a straight line.
[Note: Any alphabet from a to z can be used to represent a variable. Here we will use x, y, and z to represent variables.]
Types of Linear Equations
Depending upon the number of variables, linear equations can be of the following types:
- When the linear equation has only one variable, it is called a linear equation in one variable. For example, 2x + 1 = 4.
- If the linear equation has two variables, it is called a linear equation in two variables. For example, 3x + 2y = 4.
- If the linear equation has three variables, it is called a linear equation in three variables. For example, x – y + z = 5.
[Note: A linear equation can also have more than three variables.]
The Standard form of Linear Equations
The standard form of a linear equation in one variable is written as:
ax + b = 0, a ≠ 0
Where,
a = coefficient
b = constant
x = variable
The Solution of a Linear Equation in One Variable
The solution of a linear equation refers to the numerical value which, when substituted for the variable, satisfies the equation.
The solution of linear equations can be unique, infinite, or no solution.
For linear equations in one variable, there exists only one value that satisfies the equation. Hence, such equations always have unique solutions.
How to Find a Linear Equation with Steps?
- Transpose all the terms with variables on one side of the equation.
- Similarly, transpose all the constant terms on the other side of the equation.
- Simplify both sides of the equation and solve for the variable.
- Verify your answer.
Rapid Recall
Solved Linear Equation Examples
Example 1: If 2y – 5 = 13, what is the value of y?
Solution:
The given linear equation is:
2y – 5 = 13
⇒ 2y – 5 + 5 = 13 + 5 [Add 5 on both sides]
⇒ 2y = 18 [Simplify]
⇒ y = \( \frac{18}{2} \) [Divide by 2 on both sides]
⇒ y = 9 [Simplify]
Hence, the value of y is 9.
Example 2: If \( \frac{5}{6} \) x + 4 = 9, find the value of x.
Solution:
The given linear equation is:
\( \frac{5}{6} \) x + 4 = 9
⇒ \( \frac{5}{6} \) x + 4 – 4 = 9 – 4 [Subtract 4 from both sides]
⇒ \( \frac{5}{6} \) x = 5 [Simplify]
⇒\( \frac{5}{6} \) x × 6 = 5 × 6 [Multiply both sides by 6]
⇒ 5x = 30 [Simplify]
⇒ x = \( \frac{30}{5} \) = 6 [Divide]
Hence, the value of x is 6.
Example 3: Neon is 5 years younger than Russell. 4 years later, Russell will be twice as old as Neon. Find their present ages.
Solution:
Let Russell’s present age be x years
Then Neon’s age = x – 5
After 4 years, Russell’s age = x + 4
After 4 years, Neon’s age = x – 5 + 4
= x – 1 [Simplify]
Now, according to the question,
4 years later, Russell will be twice as old as Neon.
Therefore, x + 4 = 2(x – 1)
⇒ x + 4 = 2x – 2 [Distributive property]
⇒ 2x – x = 4 + 2 [Transpose variable terms on one side and constant terms on the other side]
⇒ x = 6 [Simplify]
So, Russell’s present age is 6 years and Neon’s present age is 1 year.
A linear equation is defined as an equation that has the highest degree of one. Linear equations are also known as equations of the first order or one degree equations.
The highest power of a linear equation is 1.
Yes, a linear equation can have one, two, or more variables.
A linear equation with one variable has one or a unique solution.