Mastering System of Equations: Definitions, Methods, and Examples

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A system of equations is a fundamental concept in mathematics that explores the interconnected relationships between multiple variables.

In algebra, a system of equations involves a set of two or more equations, each with several variables working together to describe a particular scenario. 

The study of systems of equations provides a structured approach to understanding the dynamic connections between different quantities and finding solutions that satisfy all given conditions simultaneously.

This article aims to cover the all-important concept of the system of equations.

System of Equation: Definition

The concept of a system of equations revolves around dealing with two or more equations with variables in common.

These equations can be of various types, such as linear, quadratic, polynomial, or exponential. The primary objective is to find the values for the variables that make all equations true simultaneously.

Example 1Example 2Example 3
3x + y = 56x – y = 29x + 2y = 6
6x – y = 23x – y = 32x2 – y = 1

Solving Sets of Equations:

Determining the solution of system of equations involves identifying variable values that fulfill the conditions set by each linear equation in the system.

The main purpose of solving equations is to pinpoint the variable values that make all the provided equations simultaneously true. 

These equation systems are categorized into three types based on their number of solutions.

  1. Unique (1) Solution in Linear Systems
  2. No solution in Linear System
  3. Infinity solution in the linear system

1. Unique (1) Solution in Linear Systems:

It means that there is precisely one set of values for the variables that satisfy all the equations within the system simultaneously. 

Graphically: 

This corresponds to the geometric interpretation of intersecting lines or planes at a single point and indicates the existence of a solitary solution.

No solution in Linear System:

2. No solution in Linear System:

It implies that the set of equations leads to inconsistent conditions. In this the system of equations is contradictory and there are no values for the variables that satisfy all the given conditions simultaneously.

Graphically:

This can be visualized as parallel lines or planes that do not intersect. 

3. Infinity solution in a linear system

It indicates that the equations are dependent and signify the same line or plane. In this, any point along the common line or plane satisfies all the equations and leads to an infinite set of solutions.

Graphically:

This corresponds to overlapping lines or planes. 

Methods to Solve a System of Equations:

There are several methods to solve a system of equations, each suited to different scenarios and preferences. Here are common ways to approach solving a system of equations: 

1. Substitution:

Step 1: Express one variable as a function of the other variable by solving the equation.

Step 2: Substitute that expression into the other equation and create an equation with only one variable.

Step 3: Determine the value of the leftover (remaining) variable.

Step 4: Substitute that value back into either original equation to find the value of the first variable.

2. Elimination:

Step 1: Add or subtract the equations in a way that eliminates one of the variables.

Step 2: Determine the value of the remaining variable.

Step 3: Substitute that value back into one of the original equations to find the value of the other variable.

Solved Examples of System of Equations:

These examples help us to understand how we solve the system of equations by different methods.

Example 1:  Substitution Method 

2x+y=10

3x−2y=4

Solution:

2x + y = 10 _____ (i)

3x -2y = 4 _____ (ii)

Step 1: Express one variable as a function of the other variable by solving the equation.

y = 10 – 2x

Step 2: Substitute that expression into the other (ii) equation and create an equation with only one variable.

3x -2(10 – 2x) = 4 _____ (ii)

3x -20 + 4x = 4_____ (iii)

7x -20 = 4_____ (iii)

X = 24 / 7

Step 3: Determine the value of the leftover (remaining) variable.

Put the value of x = 24/7 in (i)

2(24 /7) + y = 10

Y = 10- (48 / 7) = 22 / 7

Checking:

Put the values in any above equations to satisfy the result:

X = 24 /7, y = 22/7 

2x + y = 10 

= 2 (24 /7) + 22 /7 = 10 

Hence, our determined values are accurate.

Example 2: Elimination Method 

4x+3y=18

2x−y=5

Solution:

4x + 3 y = 18 _____ (1)

2x – y = 5______ (2)

Step 1: Add or subtract the equations in a way that eliminates one of the variables.

Before eliminating a variable, it is essential to ensure that at least one variable in both equations has the same coefficient. To achieve this, we can multiply equation (2) by -2. 

 -4x + 2y = -10______ (2*)

Add the equation (1) and modified equation (2*)

4x + 3 y = 18 _____ (1)

4x + 2y = -10______ (2*)

y         = 8 /5

Step 2: Determine the value of the remaining variable.

Substitute the previous value of y = 8 /5 in any equation to get the value of x

4X – 3(8/5) = 18_____ (1)

4x + 24 /5 = 18 _____ (1)

Add (-24)/5 to both sides:

4x = 66 /5

x = 33 /10  

So, the required result is (33 /10, 8 /5)

Final Thoughts

This article explored the fundamental concept of systems of equations, their definitions, and various methods for solving them.

We delved into the three types of solutions: unique, none, and infinite, offering insights into graphical representations.