Linear Equation

An equation is called linear when the variable’s highest power is 1. This means you won’t find squared, cubed, or high-power variables (like x2, x3…) in the equation. It includes variables raised only to the first power and does not involve any curves, powers, or products of variables. The values change at a constant rate, which makes the graph of the linear equation a straight line. The graph of a linear equation is always a straight line.

Example:

2x + 6 = 10
x = 2y
x + y = 5

Facts About Linear Equations

• The linear equation always produces a graph that forms a straight line, with no curves or bends.

• The variable’s power is always 1.

• Linear equations can have one variable, two variables, or more variables. 

• A linear equation remains balanced as long as you add, subtract, multiply, or divide both sides by the same value. 

• It is also known as a one-degree equation.

• The graph ends with a straight-line pattern. 
   

Think of an equation like a seesaw—both sides must stay balanced. In math, an equation means the left side is equal to the right side. Whatever you do on one side of the equation, do the same on the other side; otherwise, it will be unequal. We move the variable to one side and the constant to the other to simplify the equation for easier solving.  When solving an equation, you can isolate the variable by moving numbers to the other side. But when you do this, you're performing the opposite operation, so the sign or operation changes accordingly. (2x + 3 = 7 becomes 2x = 7 - 3). The +3 became -3 because you're using the opposite operation to cancel it out.

Let’s see the example,
5x - 8 = 7
 Here, we don’t know the value of x. So we move the constant from LHS to RHS, it becomes
5x = 7 + 8
Then add the constants 
5x = 15
Both 5 and 15 can be divided by 5, so we divide both sides of the equation by 5.

Dividing by 5: 5x5 = 155

Therefore, the value of x is 3.
 

A linear equation is an algebraic equation that involves variables, such as x and y, and constants, including numbers or fixed values. In a linear equation, the highest power of the variable is 1. There are several forms, namely the standard form, slope point form, slope-intercept form, etc. For example, 3x - 2y = 12 (3, 2, and 12 are the constants, and x, y are the variables).

The Standard Form

The standard form of a linear equation includes both constants and variables. In the standard form, there are different types of linear equations—some have just one variable, while others include two.

One Variable Form (Simple) 

In one variable form, an equation contains one variable. For e.g., Ax + B = 0
In this equation, A and B are constants, and x is the variable. 

Two Variable Form (Standard Form):

Ax+By+C=0
A, B, and C are constants 
x and y are variables
Both A and B cannot be 0 simultaneously.

Slope Point Form:

The slope shows how a line goes up or down as it moves across the graph. If a line has a slope m and passes through the point (x1, y1), we can write its equation in slope-point form.
y − y₁ = m(x − x₁)
You can use this form when you already know the slope of the line and one point it passes through.

Slope-Intercept Form

A common and easy way to write a linear equation is (y = mx + B), where (x) and (y) are the variables, m is the slope of the line, and B is the intercept (Intercept means the point where the line crosses the y-axis)

Example: 
y = 3x + 1 
Slope m = 3
y-intercept b = 1
 

The standard form is also known as the two-variable form. The equation contains two variables. In the Standard form, there are two different variables contained in the same equation.

Ax + By = C
A, B, and C are constants
x and y are variables

Example:
2x + 3y = 12
 

The linear equation graph is the solution that can visually show the straight line, which is why it is called linear. It shows the relationship between the x-axis and the y-axis. The line depends on the slope. When a line crosses the y-axis, that point is called the y-intercept. 
 

Example
y = 2x + 1
Take the value of x as (-1, 0, 1)
To find the value of y: 
If x is -1, y = 2(-1) + 1, y = -1
If x is 0, y = 2(0) + 1, y = 1
If x is 1, y = 2(1) + 1, y = 3
The points are (-1,-1), (0,1), (1,3)
 

A linear equation with one variable uses that variable consistently throughout the equation. 

Ax + B = 0
Ax + Bx + C = 0

Example
5x=20
x = 4
5(4) = 20
The value of x is 4
 

In the equation, there are two different variables. Both variables have a degree of 1. For example, the equation (5x + 2y + 8 = 0) has variables (x) and (y), and the numbers 5, 2, and 8 are constants.
This equation contains a straight line.

For example
2x + 3y = 12
 In this equation, we want to find the value of x and y
Let’s take the value of x is 0
 then 2(0) + 3y = 12
 3y = 12
 y = 12/3 
 = 4 (The 3 is multiplying the variable on the left-hand side (LHS). To isolate y, we divide both sides by 3.)
y = 4
The value of x and y is (0,4).
 

Think of a linear equation as a scale: both sides must stay the same to stay balanced. 
We can do the same thing on both sides (LHS, RHS) so the balance is not disturbed.

Step 1: Combine the equations in simplified form for better understanding

Step 2: Change the variable from one side to the other side of the equation

Step 3: Solve the equation

Example 
Solve the Linear equation 3x-2=4
Solution: 
Change the constant LHS to RHS
3x - 2 = 4
Add 2 on both sides of the equation
3x = 4 + 2
3x = 6
 When the value 3 is moved from the left-hand side (LHS) to the right-hand side(RHS), it becomes a divisor because we’re dividing both sides by 3. 
x = 63
After dividing the value, you will get the value of x
x = 2.
 

Linear equations help in real life, not just in school. They’re useful for budgeting, cooking, and improving daily routines.

Budgeting

Linear equations help make a budget to save money. For example, Ajay wants to buy a Lego set that costs $1000.
He has already saved  $200 in his piggy bank.
He earns $50 each day by helping his mother.
Find out how many days it will take for Ajay to save enough money.

Let x be the number of days he works.
Each day, he earns $50, so his total savings after x days are:
Total Money = 50x + 200
We set up the equation:
50x + 200 = 1000
Now solve the equation:
50x = 1000 − 200
50x = 800
 x = 800 ÷ 50
 x = 16
So, Ajay needs to work for 16 days to buy the Lego set.

Cooking: 
You’re cooking biryani for x people. It takes 2 cups of rice to serve 4 people. How many cups of rice will you need for x servings?

2 cups of rice make 4 servings.
How much for x servings?
Linear Equation:
Rice needed = 24x
                x  = 0.5 

Traveling:
When you're going on a trip and want to reach your destination on time, a linear equation can help. For example, if you're driving at a speed of 60 kilometers per hour, the distance you travel depends on how long you drive. Distance = 60 
So, x = 60.