Why the Substitution Method Works for Solving Systems of Linear Equations

The substitution method is a powerful tool in algebra for solving systems of linear equations. It is based on several fundamental principles and properties of equality, making it a reliable and efficient method. This article will delve into the mechanics and underlying concepts of this method.

Basic Principle of Equality

The core of the substitution method lies in the basic principle of equality. This principle asserts that if two expressions are equal, you can replace one with the other in any equation. This principle is the backbone of the substitution method, allowing for the transformation of complex systems into simpler forms.

Reduction of Variables

One of the primary advantages of the substitution method is its ability to reduce the number of variables. By isolating one variable in one equation, you can express it in terms of the other variable(s). This reduction simplifies the system, turning a set of multiple equations into a single equation with a single variable, making it much easier to solve.

Stepwise Solution

The substitution method proceeds in steps. Once you have isolated a variable, you substitute its expression into the other equations. This substitution simplifies the equations, allowing you to solve for the isolated variable. After finding the value of one variable, you substitute it back into one of the original equations to find the value of the other variable. This step-by-step process ensures a clear and logical path to the solution.

Consistency and Existence

The method works on the assumption that the system is consistent, meaning it has at least one solution. If the equations represent lines in a plane, the substitution method will find the point of intersection, provided one exists. If the lines are parallel, there will be no solution, and if they coincide, there will be infinitely many solutions.

Example Walkthrough

Let's consider a practical example to illustrate the application of the substitution method.

Consider the following system of linear equations:

Equation 1:

Equation 2:

Step 1: Substitute the expression for y from the first equation into the second equation.

3x - (2x - 3) 12

Step 2: Simplify the equation.

3x - 2x 3 12

Step 3: Solve for x.

x 3 12

5x 9
x 9/5

Step 4: Substitute x 9/5 back into the original equation to find y.

y 2(9/5) - 3

y 18/5 - 15/5

y 33/5

The solution to the system is (9/5, 33/5).

Conclusion

The substitution method is effective because it transforms the system into a simpler form that can be solved step by step. By leveraging the basic principle of equality and the reduction of variables, it ensures a systematic and logical approach to solving systems of linear equations. This method is particularly useful when one equation can be easily manipulated to express one variable in terms of another.