Linear Equations

A linear equation is an algebraic equation in which the highest exponent of the variable is 1.

Here is an example of a linear equation:

7x + y = 8

This is the standard form of writing a linear equation. 

The graph of a linear equation always gives a straight line, as shown:

We can also write this equation in some other ways:

• y = -7x + 8 (the Slope-Intercept Form)
• y + 6 = -7(x – 2) (the Point-Slope Form)

However, these are NOT linear equations:

• y = 5x² – 2 (as the exponent of the variable x is 2)
• x – y³ = 4 (as the exponent of the variable y is 3)
• ${\sqrt{x}-y}$ = ${3}$ (as the exponent of the variable x is ${\dfrac{1}{2}}$)

Formulas

Linear equations can be grouped based on the number of variables.

In One Variable

The standard or general form of a linear equation in one variable is:

Ax + B = 0

Here,

• A and B are the constants
• x is the variable

In Two Variables

The standard form of a linear equation in two variables is:

Ax + By = C

Here, 

• A, B, and C are the constants
• x and y are the variables 

In Three Variables

The standard form of a linear equation in three variables is:

Ax + By + Cz = D

Here, 

• A, B, C, and D are the constants
• x, y, and z are the variables

System of Linear Equations

A system of linear equations contains two or more linear equations that involve the same set of variables. The solution to a system of linear equations is the set of values of the variables that satisfy all the equations simultaneously.

For example,

2x + 3y = 12 and x – y = 4 both are linear equations. Together, they form a system of linear equations. 

Solving

For One-Variable

When solving equations involving one variable, we first isolate the variable to one side and then solve for the variable. 

Let us solve 2x + 5 = 11

Subtracting 5 from both sides, we get

⇒ 2x + 5 – 5 = 11 – 5

⇒ 2x = 6

On dividing both sides by 2, we get

⇒ ${\dfrac{2x}{2}=\dfrac{6}{2}}$

⇒ x = 3

Thus, the variable x = 3

For Two-Variables

The two-variable linear equations can be solved either by substitution or elimination method:

1. Substitution Method
2. Elimination Method

Note: Linear equations in two variables can also be solved graphically, by cross-multiplication, or by the determinant method.

Graphing

The graph of a linear equation with one variable, x, forms a vertical line parallel to the y-axis.

Similarly, the graph of a linear equation with one variable, y, forms a horizontal line parallel to the x-axis.

The graph of a linear equation of two variables, x and y, always gives a straight line.

This is because a linear equation represents a constant rate of change between the two variables. 

• A horizontal line has the equation y = c, with m = 0
• A vertical line has the equation x = a with an undefined slope

Let us graph the linear equation y = 2x + 3

Step 1: Rewriting in Slope-Intercept Form (If Necessary)

As we know, the slope-intercept form is y = mx + c

The given equation, y = 2x + 3, is already in slope-intercept form.

Here,

• m = 2 
• c = 3 

Step 2: Identifying the Slope and Y-Intercept

Y-Intercept (c): This is the point where the line crosses the y-axis. In this case, c = 3. Thus, the y-intercept is (0, 3)

Slope (m): The slope m = 2 means that if x increases 1 unit, y increases by 2 units (rise = 2, run = 1)

Step 3: Using the Slope to Find Another Point

Using the y-intercept and the slope, here are some more values:

x	2x + 3	y
-2	2(-2) + 3	-1
-1	2(-1) + 3	1
0	2(0) + 3	3
1	2(1) + 3	5
2	2(2) + 3	7

Step 4: Plotting the Points on the Coordinate Plane