Linear Equations for Beginners: Simple Explanation with Examples 

Introduction 

Lets suppose you are sitting in a aptitude test you see a question like “ You have ten apples and shear equally these apples among your friends in this way each friend get 2 apples” or may be you see such kind expression like “2x + 3x +9 = 5x + 1” how could you solve this?

If you are facing a problem to solve these questions you are not the only one. There are a lot of students even college students who are also facing these problems and sometimes gives the excuse as “Math makes my brain hurt, I don't like maths,  I failed algebra in school, all those variables and number confuse me” because of many reasons like weak base of algebra, having not interest in maths or my be sometimes slackness of teachers.

In this blog post, I will discuss step by step how to solve a linear equation. How to solve algebraic word problems? What is the meaning of linear  equation and real world application of linear equation in the way by the end of this guide will be able to solve linear equation.

What Linear Equations Really Are

Linear means straight line. So, those equations give a straight line on the graph; it will be a linear equation because it represents a constant rate of change. 

In an easy manner it can be defined as those equations having maximum power 1 of variable are called linear equations.

Equation Represention

• Standard Form:

ax + by = c

Where a,b and c are constant and x and y are variable 

• Slop Intercept Form:

y = mx + b

Where x and y variable, m is slope and b is intercept 

Visual Recognition Of Linear Equations Linear Equations look like in the wild:

• Simple: x + 3 = 7

One step equation and all values  are positive

• Medium: 2x - 5 = 9

Two step equation with one negative value

• Trickier: 4x + 1 = 3x + 6

Both sides having variable and constants 

The "One Power Rule"

The above equation shows that linear equation always having maximum power 1. 

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The Universal Solving Formula 

For solving linear equations either the one step, two step or multistep STAR method are the best method for solving linear equations 

Before learning the method first we need to understand what STAR method is.

The STAR Method (Our Secret Weapon)

STAR method is the abbreviation of:

S - Simplify both sides
T - Track down the variable
A - Apply inverse operations
R - Reality check your answer

Lets Apply STAR method. Let's consider an equation 3x+7=22 .

Example: 3x + 7 = 22

Step S: Simplify

Nothing to simplify here because both sides are already simplified. One side has one variable and constant and the other side has constant.

Step T: Track the Variable

Now analyse the position of variable as x is being multiplied by 3, then 7 is added to the result

Step A: Apply Inverse Operations

In order to make variable “x” alone 

• Undo the +7 by subtracting 7 from both sides as a result we get 

3x = 15

• Undo the ×3 by dividing both sides by 3 as a result we get the answer as

x = 5

Step R: Reality Check

In the last step we should check the answer. For this purpose we will put the answer value of x which is equal to “5’ in the question

3(5) + 7 =22

15 + 7 =22

22 = 22

Hence if both sides of the equation are equal then the value of x =5 is true for this equation.

The Golden Rules That Never Fail

1. Whatever you do to one side, do to the other

In order to balance the equality in the equation apply the same operation on both sides of the equation 

Example: x + 3 = 4

Apply operation as -3 on both sides of the equation

X + 3 - 3 = 4 - 3

X = 1.

2. Isolate the variable

In order to get the answer make variable alone by reverse operation 

Example: 2x + 4 = 6

First add -4 on both sides of equation

2x + 4 - 4 = 6 - 4

2x = 2

Now make x alone by dividing 2 on both sides of equation

2x/2 = 2

X = 1

3. Always check your answer by substituting back

In order to check answer is true for the given answer put the value of x in main equation 

Example: 2x + 4 = 6

This is the above equation and for this equation we get the answer x = 1. For the reality check put x = 1 in the main question at the place of variable.

2(1) + 4 = 6

2 + 4 = 6

6 = 6

Hence if both sides of the equation are equal then the value of x=1 is true for the given equation.

When Variables Appear Twice

Lets take another example of linear equation when variable appear on the both sides of equation 

Example: 4x - 3 = 2x + 9

• First collect variable one side of equation and the constant on the other side of equation but the variable always keep positive 

For this purpose, move 2x from one side to the other side of the equation and it becomes -2x and -3 will move from one side to the other side of the equation and it becomes +3. 

4x - 2x = 9 + 3

• Simplify both sides of equation

2x = 12

X = 12/2

X = 6

•  Now, check the answer by putting 6 at the place of x in given question

4(6) - 3 = 2(6) + 9

24 - 3 = 12 + 9

21 = 21

General Map For Solving Linear Equation

Real-World Problem Solving Made Simple 

Example 1: The Rideshare Dilemma

Problem: A car  charges $5 upfront plus $1per mile. Your trip cost $10. How many miles did you travel?"

• Convert problem into equation:

 5 + 1x = 10

• Now solve the equation by add - 5 on both sides of equation

5 + x - 5 = 10 - 5

• Simplify 

X = 5

The answer is 5 miles.

Example 2: The Gym Membership Mystery

Problem: "Planet Fitness charges $10 monthly plus a one-time $50 signup fee. You paid $130 total. How many months were you a member?"

• Convert into Equation setup

 50 + 10x = 130

• Add -50 on both sides of equation

50 + 10x -50 = 130 - 50

10x = 80

X = 80/10

X = 8

You will be 8 months member

Level-Up Challenges with Full Solutions 

Beginner Challenges (Build Confidence)

These problems are related to single operations like addition, subtraction, multiplication or division. These problems can be solved in one step. That's why these equations are called one step equations.. These equations are very helpful for a beginner to build his confidence. Lets solve some examples

1. x - 12 = 8 

Add 12 on both sides of equation

X -12 +12 = 8 + 12

Simplify both sides of equation 

X = 20

2. 4x = 28 

Divide both side of equation by 4

4x/4 =28/4

X = 7

3. x + 15 = 23 

Subtract both side of equation by 15

X + 15 - 15 = 23 - 15

X = 8

Intermediate Challenges (Apply Skills)

These equations involve more than two operations that's why these equations are called two step equations.

1. 3x + 5 = 20 

Move 5 on the other side of equation

3x = 20 - 5

3x = 15

Divide both side of equation by 3

X = 5

2. 2x - 7 = x + 4 

Move variable on the one side of equation and constant move on the other side of equation

2x - x = 4 + 7

X = 11

3. 6 + 4x = 2x + 18 

4x - 2x = 18 - 6

2x = 12

X = 6

I hope this linear equation tutorial for beginners  will be helpful to understand what is a linear equation simple explanation and you will be able to solve algebra equation step by step which is very helpful to solve one variable equation which is a easiest way to solve linear equation without fear which is also an algebra help for adults.