Linear Equations

Introduction

In this article, we will introduce the linear equations from Mathematics for the IIT JEE Mains and advanced exams. This topic educates the learners on linear equations. Here, we will discuss the linear equations, forms of linear equations, steps to solve a linear equation, hints in linear equations, and linear equations in one variable. This article will also help you to understand the basics of linear equations.

Algebra is a part of mathematics for IIT JEE mains and advanced exam preparation that requires a lot of practice. A linear equation is an integral part of algebra in mathematics.

Let us start with defining the linear equations 

What are Linear Equations?

An equation with the highest degree of power 1 of a variable is called a linear equation.

It means that in a linear equation there is no variable that has an exponent of more than 1. A linear equation graph always forms a straight line. This is the reason it is named as a linear equation. A linear equation in which only one variable is present is of form Mx + N=0, where M and N are constant and x is a variable.  A linear equation in which two variables are present is of form    Mx + Ny = P, where M, N and P are constant and x and y are variables.

What are the Different Forms of Linear Equations?

There are three forms of equations. These are as follows:

1. Standard form of linear equation
2. Point slope linear equation
3. Slope-intercept linear equation

What is the Standard Form of a Linear Equation?

The standard form of linear equations can be written in three forms i.e. in one variable, two variables, and three variables. It is also called a general form of linear equation.

1. In one variable: px + q = 0, where p and q ≠ 0 and integer. x is a variable.
2. In two variables: px + qy = r, where p, q, and r ≠ 0 and are integers. x and y are variables.
3. In three variables:  px + qy + rz =  s, where p, q, r, and s ≠ 0 and are integers. x, y, and z are variables.

Rules for Standard Form of Linear Equations:

• Must have the form Px + Qy = R
• P, Q, and R must be integers
• P cannot be negative
• P, Q, and R should have no common factors other than 1

Point Slope Form of a Linear Equation:

A straight line equation is formed by considering the points in the x-y plane, such that: y – y1 = m(x – x1) is a point-slope form of a linear equation.

In this equation, x1 and y1 are the coordinates of the point and m is the slope of the line.

Slope-Intercept Form of a Linear Equation:

It is the most common form of a linear equation that is represented as 

y = mx + c, where

m is the slope of the line,

c is the y-intercept

y and x are the variables of the y-axis and x-axis, respectively.

For example, y = 3x + 7:

slope, m = 3 and intercept = 7

If a straight line is parallel to the x-axis, then the x-coordinate will be equal to zero. Therefore,

y = c

If the line is parallel to the y-axis then the y-coordinate will be zero.

mx + c = 0

x = -c/m

Linear Equation in One Variable:

Linear equations in one variable are the equations where there is only one variable and which exponent value is 1.

Steps to solve a linear equation:

1. The first step is to simplify each side if required.
2. The next step is to use addition or subtraction properties to move the variable term to one side and all other terms to the other side.
3. The next step is to use multiplication or division in a linear equation
4. The last step is to check your solution.

Now, let’s solve an example following the above-mentioned steps.

Example:  Solve 6x – 32 = 8 – 2x

Solution:

Linear equation: 6x – 32 = 8 – 2x

Step 1: transfer variables to the one side of the equation

  6x + 2x = 8 + 32

Step 2: solve by adding or subtracting:

6x + 2x = 8 + 32

8x = 40

Step 3: divide the equation with 8 into both sides

8x = 40

8x/8 = 40/8

x = 5

Step 4: verify the answer by putting the value of x in a linear equation:  

6x – 32 = 8 – 2x

6*5 – 32 = 8 – 2*5

30 – 32 = 8 – 10

-2 = -2

The left-hand side is equal to the right-hand side, hence the value of the x that we get solving the equation is correct.

Hints in Linear Equations:

• The value of the variable that generates a valid linear equation is named the solution or root of the linear equation
• The outcome of a linear equation is unchanged if the same number is subtracted, added, divided, or multiplied into both sides of the equation
• The graph of a linear equation in one or two variables constantly forms a straight line

Conclusion

An equation that has a variable with the highest degree of 1 is known as a linear equation. It has three forms: The Standard form of linear equation, Point-slope linear equation, and Slope-intercept linear equation. The standard form of the linear equation can be in one variable, two variables, and three variables but the exponent value of variables will always be 1.