Linear Equations

A linear equation is an integral part of algebra in mathematics. An equation with a variable with the highest degree of one is known as a linear equation. It has three forms: A standard linear equation, Point slope linear equation, and slope-intercept linear equation. The Standard Form of the linear equation can be in one variable, two, and three variables, but the exponent value of variables will always be one.

This article talks about linear equations. You will find brief information on the concept of linear equation induction in Maths, a thorough explanation of different types of linear equations, rules of a linear equation and so on. So, let’s start by describing the linear equation in the Maths study material.

What are Linear Equations?

A linear equation can be defined as an equation that has the highest degree of power of one variable. In other words, no variable has an exponent of more than one in a linear equation. In a graph, a linear equation constantly forms a straight line. Hence, it is named the linear equation. 

• A linear equation in which only one variable is present, for example, Mx+N=0, where M and N are constant, and x is a variable.  
• A linear equation in which two variables are present, for example, Mx+Ny=O, where M, N, and O 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:

• A standard form of linear equation
• Point slope linear equation
• Slope intercept linear equation

What is the standard form of a linear equation?

There are three ways in which the standard form of linear equations can be written. These include one variable, two variables, and three variables, and 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 are integers. x and y are variables.
3. In three variables: px+qy+rz=s, where p, q, r, and s ≠ 0 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 linear equation

The straight-line equation is formed when the points are considered in the x-y plane, in a way that y-y₁=m(x-x₁) is a point-slope form of a linear equation.

In this equation, y1 and x1 are the coordinates of the point. 

Slope intercept form of linear equation

It is the most common form of a linear equation that is represented as x = my + a, where

m is the slope of the line,

a is the x-intercept

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

For example, x = 3y + 7:

slope, m = 3 and intercept = 7

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

y=b

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

mx+b = 0

x=-b/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

Here are steps to solve the 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 outcome of a linear equation cannot be changed if the same number is subtracted, added, divided, or multiplied into both sides of the equation.
• The value of the variable generating a linear equation valid is named the solution or root of the linear equation. 
• The graph of a linear equation in one or two variables constantly forms a straight line.

Conclusion

With this, we come to an end to linear equations. We studied, in a linear equation, there is no variable that has an exponent of more than one. The linear equation always plots a straight line on the graph. 

In this article describing linear equations, we studied the concept of linear equations in length. We covered several other topics, such as the rules of linear equations, steps to solve linear equations, and other related topics. We hope this study material must have helped you better understand linear equations.