Introduction to Solving Systems of Linear Equations

In mathematics, particularly in algebra, understanding how to solve systems of linear equations is crucial for many applications. A system of linear equations involves multiple equations that are linear in nature and contain the same set of variables. This tutorial will explore the solution of a specific system of linear equations using both the elimination and Gaussian pivot methods.

The System of Equations

Consider the following system of equations:

-212y 24 (Equation I) -5x 3y 6 (Equation II)

Solving the System Using the Elimination Method

The elimination method is a widely used technique to solve systems of linear equations. It involves manipulating the equations to eliminate one of the variables, thereby simplifying the problem.

Step 1: Simplify the Second Equation

First, we can multiply the second equation by 4 to align the coefficients of (y) with the first equation:

-212y 24 -2 12y 24

Now we have:

-212y 24 (Equation I) -2 12y 24 (Equation II, after simplification)

Step 2: Compare the Equations

Notice that both equations are now:

-212y 24 -2 12y 24

This indicates that they are identical, meaning they are just different representations of the same line. Now, let's solve one equation for a variable:

-212y 24 12y 24 - 212y y (24 - 212y) / 12 y 2 / 12 5 / 3x - 2 y 5 / 3x - 2

This result tells us that there are infinitely many solutions because the equations describe the same line. Any point ((x, y)) that satisfies the equation of this line is a solution.

Solving the System Using Gaussian Pivot

Let's also solve the system using the Gaussian pivot method. Follow these steps:

Step 1: Represent the System in Matrix Form

[begin{align} begin{cases} -212y 24 [text{I}] -5x 3y 6 [text{II}]end{cases} Leftrightarrow begin{cases} -212y - 2 - 12y 24 - 24 [text{III}text{I}-4text{II}] y frac{5}{3}x - 2 [text{II}]end{cases} end{align}end{p>

Now, the system is reduced to:

[begin{cases} 0 0 [text{III}] y frac{5}{3}x - 2 [text{II}]end{cases}end{p>

Since the third equation simplifies to 0 0, it means that we have an identity, confirming that the two original equations are dependent.

Interpreting the Solutions

When both equations are identical, the system has an infinite number of solutions. This means that for any real number (x), there exists a corresponding (y) that satisfies the equation (y frac{5}{3}x - 2).

Thus, the solution set can be expressed as:

[boxed{x, y x frac{5}{3}x - 2 text{ for any } x in mathbb{R}}end{p>

Conclusion

In summary, we've demonstrated two methods to solve the system of linear equations: the elimination method and the Gaussian pivot method. Both methods confirm that the system has an infinite number of solutions, meaning there are infinitely many points ((x, y)) that lie on the line defined by the equation (y frac{5}{3}x - 2).

Understanding how to identify and solve such systems is an essential skill in algebra, with applications in various fields such as physics, engineering, and economics. By mastering these techniques, you can effectively handle more complex problems in mathematics and real-world scenarios.