Dependent system of linear equations is a set of equations in which all the equations represent the same line when graphed. This means that the system has an infinite number of solutions, as every point on the line satisfies all the equations simultaneously. Essentially, in a dependent system, the equations are different forms of the same equation, possibly scaled or rearranged.

Both equations represent the same line because the second equation is simply the first equation divided by 2. Graphically, these equations will coincide, meaning they lie on top of each other. Hence, any point that lies on this line will be a solution to both equations.

What is Linear Equation?

A linear equation is a mathematical equation that forms a straight line when graphed on a coordinate plane. It can be written in various forms, but the most common is the slope-intercept form:

y = mx + b

Here, y and x are variables, mmm represents the slope of the line (which indicates the steepness and direction), and b is the y-intercept (the point where the line crosses the y-axis).

System of Linear Equation

A system of linear equations is a collection of two or more linear equations involving the same set of variables. The goal in solving a system of linear equations is to find values for the variables that satisfy all the equations simultaneously. Systems of linear equations if consistent can be classified into two categories:

• Dependent System of Linear Equation
• Independent System of Linear Equation

Dependent System of Linear Equation

A dependent system of linear equations is a set of equations that have an infinite number of solutions.

This occurs when the equations in the system are essentially the same, meaning they represent the same line when graphed. Therefore, any point that lies on this line will be a solution to all the equations in the system.

Examples of Dependent Systems of Linear Equations

Some example of dependent system of linear equations are:

• 2x + 4y = 8
• x + 2y = 4

• 3x- 6y = 12
• x- 2y = 4

• 4x + 2y = 10
• 2x + y =5

• 2x + 4y + 6z = 12
• x + 2y + 3z = 6
• 4x + 8y + 12z = 24

Methods of Solving Systems of Linear Equations

There are several methods to solve systems of linear equations, including:

• Graphical Method
• Substitution Method
• Elimination (or Addition) Method
• Matrix Method (using Gaussian Elimination or Inverse Matrices)

How to Identify a Dependent System of Linear Equations?

For a system of two linear equations in the form:

• a₁​x + b₁​y = c₁​
• a₂x + b₂y = c₂

The system is dependent if:

[Tex]\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}[/Tex]

For a system of three linear equations in the form:

• a₁​x + b₁​y + c₁​z = d₁​
• a₂x + b₂y + c₂z = d₂
• ​a₃x + b₃y + c₃z = d₃

The system is dependent if:

[Tex]\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = \frac{d_1}{d_2}[/Tex]

and

[Tex]\frac{a_1}{a_3} = \frac{b_1}{b_3} = \frac{c_1}{c_3} = \frac{d_1}{d_3}[/Tex]

Dependent Vs Independent System of Linear Equation

Common differences between dependent and independent system of linear equations are listed in the following table: