Solving Systems of Linear Equations: A Comprehensive Guide

In mathematics, solving systems of linear equations involves finding the values of variables that satisfy multiple equations simultaneously. This article will explore a step-by-step approach to solving the system of linear equations:

Introduction to the Problem

The system of equations given is:

2x y 1

3x - y 4

We can use either the substitution method or the elimination method. In this article, we will demonstrate both methods and provide a graphical solution to validate the results.

Solving the System Using the Elimination Method

Let's start by demonstrating the elimination method:

Step 1: Add Both Equations

First, we add both equations to eliminate the y variable:

2x y 3x - y 1 4

Combining like terms:

5x 5

Now, we can solve for x:

x 1

Step 2: Substitute x Back into One of the Original Equations

We substitute x 1 into Equation 1 to find y:

2(1) y 1

2 - y 1

Subtracting 2 from both sides:

y 1 - 2 -1

Final Solution

The solution to the system of equations is:

x 1

y -1

Verification

To verify, we substitute x 1 and y -1 back into both original equations:

For Equation 1: 2(1) - (-1) 2 1 1 (True)

For Equation 2: 3(1) - (-1) 3 1 4 (True)

Both equations are satisfied, confirming the solution.

Solving Using the Substitution Method

Alternatively, we can solve the system using the substitution method:

Step 1: Solve One Equation for One Variable

Let's solve Equation 1 for y:

2x y 1

y 2x - 1

Step 2: Substitute into the Other Equation

Substitute y 2x - 1 into Equation 2:

3x - (2x - 1) 4

3x - 2x 1 4

x 1 4

x 4 - 1 3

Now, solve for y using the value of x:

y 2(3) - 1

y 6 - 1

y 5 - 1 -1

Final Solution

The solution using the substitution method is:

x 1

y -1

Graphical Solution

In a graphical solution, the lines representing these equations will intersect at a single point, which gives the solution. The y-terms have the same coefficients but opposite signs, allowing us to eliminate y by adding or subtracting the equations.

By adding the two equations:

2x y 3x y 1 4

Combines to:

5x 5

Solving for x gives:

x 1

Substitute x 1 back into the first equation:

2(1) y 1

2 y 1

Solving for y gives:

y 2 - 1 -1

Thus, the solution to the system of equations is:

x 1

y -1

Conclusion

Both the elimination and substitution methods provide a systematic approach to solving systems of linear equations. The graphical method offers a visual representation, confirming the algebraic results.

Additional Resources and Keywords

This article covers the following key concepts:

System of equations: A set of two or more equations with the same variables.

Linear equations: Equations where the highest power of the variables is one.

Substitution method: Solve one variable in terms of another and substitute into the other equation.

Elimination method: Add or subtract equations to eliminate one variable.