Algebraic method of solving a pair of linear equations

Solving Linear Equation Systems with Two Variables

A pair of linear equations in two variables can be solved using different techniques. These methods help find the point(s) where both equations are satisfied. Commonly used methods include:

• Algebraic Method
• Graphical Method
• Matrix Method
• Determinant Method (Cramer's Rule)

1. Algebraic Method

Algebraic methods use mathematical operations to eliminate variables and solve the system. They are accurate and useful for most equation pairs.

Types of Algebraic Methods:

• Substitution Method – Solve one equation for one variable and substitute it into the other.
• Elimination Method – Add or subtract equations to eliminate one variable.
• Compression Method (Reduction) – Simplify or scale both equations to make them comparable, then eliminate a variable.
• Cross-Multiplication Method – Eliminate variables using cross-multiplication when equations are in standard form.

Compression Method (Reduction Method)

This method involves simplifying both equations (by division or multiplication) so they can be directly subtracted or added to eliminate a variable.

Example:

3x + 2y = 12   ...(1)
6x + 4y = 24   ...(2)
  

Divide equation (2) by 2:

⇒ 3x + 2y = 12   ...(1)
   3x + 2y = 12   ...(2 simplified)
  

Subtract (2) from (1):

(3x + 2y) - (3x + 2y) = 0  
⇒ 0 = 0 (This indicates infinitely many solutions, as both equations represent the same line and are dependent.)

2. Graphical Method

Plot both equations on a graph. The point of intersection gives the solution.

• Unique Solution: The two lines cross at exactly one point, meaning the system is consistent and independent.
• No Solution: The lines never intersect because they are parallel, indicating an inconsistent system.
• Infinite Solutions: Lines coincide (dependent and consistent).

3. Matrix Method

The system is written as AX = B, where:

• A is the coefficient matrix
• X is the variable matrix
• B is the constant matrix

The solution is found using: X = A^-1B, if A is invertible.

4. Determinant Method (Cramer's Rule)

This method uses determinants to solve systems of equations in the form:

a₁x + b₁y = c₁  
a₂x + b₂y = c₂  
  

Steps:

• Calculate the primary determinant (D) using the coefficients of x and y.
• Substitute the respective columns with constant terms to compute D_x and D_y.

Then solve using:

x = D_x / D, y = D_y / D

Detailed Explanation: Substitution Method

The substitution method solves one equation for one variable and substitutes it into the second equation.

Steps:

1. Solve one equation for one variable.
2. Substitute into the second equation.
3. Simplify and solve for the second variable.
4. Substitute back to find the first variable.

Example:

x + y = 5   ...(1)  
2x - y = 4  ...(2)
  

Step 1: Solve equation (1) for y:

y = 5 - x

Step 2: Substitute y in equation (2):

2x - (5 - x) = 4  
2x - 5 + x = 4  
3x = 9 ⇒ x = 3
  

Step 3: Substitute x = 3 into equation (1):

y = 5 - 3 = 2

Final Answer: x = 3, y = 2

Advantages of the Substitution Method:

• Easy to understand
• Works well when one variable is already isolated

Limitations:

• May become time-consuming when dealing with complex values or fractional coefficients.
• Not ideal for systems with awkward coefficients

Each method has its own advantages. Algebraic methods like substitution, elimination, and compression are commonly used for manual solving. Graphical methods offer visual clarity, while matrix and determinant methods are efficient for larger or complex systems. Mastering all methods strengthens your overall understanding and problem-solving ability in algebra.