“Linear Equation” is an important concept of mathematics, which is an evident part of scholastic as well as the syllabus of competitive exams. The advanced level study of the topic for the unravels that one can solve the linear equation using one variable or two variables. Questions on linear equations in both one and two variables are often seen in the entrance for competitive exams. Thus, if you are gearing up for competitive exams like SSC CGL, RBI, etc. or scholarships exams like MPTAAS, NTSE, etc., then, hair we are to help you with simplified notes on this topic straight.
A concept that holds an important position in mathematics is linear equations. Are you confused about what they are and how to solve them? Do the variables all get jumbled when you try to solve them?
The linear equation is a complex concept of algebra that involves specific rules and principles that are required to solve the questions and derive the value of the variable. The topic of linear equation is an important part of the NCERT school curriculum, and those wishing to excel and get a high score and wondering about CGPA calculators and how to pull up their scores should focus on linear equations and practice hard to master the concept.
Meritorious students facing financial constraints can also hope to be awarded a scholarship like MPTAAS by working on their score and proving their academic acumen. Mentioned below is a comprehensive and extensive guide to linear equations, the multiple concepts associated with it, and notably the steps involved to solve a linear equation.
Linear Equation: Linear equation, as can be understood from the term, refers to the equation used to represent a straight line. A linear equation having two variables and constant(s) is known as a Linear Equation in Two Variables, whereas, when it has one variable, it is called a Linear Equation with One Variable. However, the point to be remembered is that variables in the linear equations can’t have exponents or be under roots.
General Form of Writing a Linear Equation
Although there are numerous ways of writing a linear equation in one variable, mentioned below is the general standardized way of writing it.
ax+b=c, where ‘a’ and ‘b’ are constants and ‘x’ is the variable
The general method of writing a linear equation in two variable is as follow:
ax+by=c, where ‘a’, ‘b’ and ‘c’ are constants and ‘x’ is the variable
Principles to be remembered when solving a linear equation:
To solve a linear equation, certain principles have to be kept in mind to process ahead with it. These principles help in facilitating the steps of getting the value of the variable. These principles are embedded in the basic understanding of maths and are heavily used in solving linear equations. Enlisted below are such facts that are essential to be known to solve a linear equation.
- If the same number is added to both the sides of an equation, it doesn’t bring about any change in the equation, i.e.
If a=b, then a+c=b+c
- If the same number is subtracted from both the sides of an equation, it doesn’t affect any change in the equation, i.e.
If a=b, then a-c=b-c
- Similar to the above-mentioned points, multiplying both sides of an equation with the same number results in both the sides remaining equal to each other, i.e.
If a=b, then ac=bc
- If both the sides of an equation are divided by the exact same, non-zero number, then the two sides of the equation remain equal, and the equation remains unchanged i.e.
If a=b, then a/c=b/c where c is a non-zero number
Process of Solving a Linear Equation in One Variable
The process of solving a linear equation involves multiple steps that are to be followed simultaneously. The equality sign in the middle of the twos ides denotes that the two sides of the equation are balanced. To solve an equation, it is imperative for both sides to be balanced. Mentioned below are the steps to be followed when solving a linear equation in one variable.
- In the case of an equation with a fraction, clear the fraction by making the denominator the same and changing it into an equation without any denominator.
- Following this, simplify the fraction by using the properties of addition, subtraction, multiplication, and division. Ensure that the changes made are balanced on both sides in order to avoid affecting the balance of the equation.
- After this, isolate the variable ‘x’ on one side and take all the constants on the other side of the equation and solve for ‘x’ to get the value of the variable.
Examples of Linear Equation in One Variable
Here are some examples that will clearly depict how to solve a linear equation in one variable by a step-by-step process.
Eg1: Solve X when X=12(X+2)
Step1- Simplify by removing the bracket:
X=12(X+2)
= X=12X+24
Step2- Subtract 24 from each side:
X-24=12X+24-24
= X-24=12X
Step3- Simplify by isolating ‘X’:
X-12X=24
= -11X=24
Step4- Isolate ‘X’ by just keeping it on one side:
X=-24/11
Thus the value of the variable X is -24/11.
Eg2: Solve for x when 2x-5=3(x-1)
Step1- Simplify by removing the bracket:
2x-5=3(x-1)
= 2x-5=3x-3
Step2- Bring the variables on one side and the constants on one side:
2x-3x=-3+5
= -x= 2
= x=2
Thus the value for x is 2
Ways of Solving Linear Equation in Two Variables
- In linear equation with two variables, there are a set of two equations that are together called a system of equations, and both the equations are used to solve for the value of the two variables. The different methods used for solving the linear equations in two variables are listed below.
- Substitution Method
- Method of Elimination
- Method of cross-multiplication
- Determinant Method
The solutions will vary according to questions but this is the major process of solving a linear equation in one variable. Relentless practice can help a person excel in the concept, and with practice, it will be easier to handle more advanced Math and algebra questions of linear equations that need a better understanding and a strong foundation.
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