Solving 2x + 8y = 6 And -5x - 20y = -15 A Detailed Explanation

In the realm of mathematics, particularly in algebra, solving systems of linear equations is a fundamental skill. These systems often represent real-world scenarios, from balancing chemical equations to optimizing business strategies. This article delves into the intricacies of solving the given system of linear equations:

2x + 8y = 6
-5x - 20y = -15

We will explore different methods to solve this system, analyze the nature of the solution, and discuss the implications of the results. Understanding these concepts is crucial for anyone venturing further into mathematics, engineering, economics, or any field that relies on mathematical modeling.

Methods for Solving Systems of Linear Equations

There are several methods to tackle systems of linear equations, each with its strengths and weaknesses. The most common methods include:

• Substitution: This method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be easily solved.
• Elimination (or Addition/Subtraction): This method focuses on manipulating the equations so that when they are added or subtracted, one of the variables is eliminated. This again leads to a single equation with one variable.
• Graphical Method: This method involves plotting the equations on a graph and finding the point(s) where the lines intersect. The coordinates of the intersection point(s) represent the solution(s) to the system.
• Matrix Methods: For larger systems of equations, matrix methods like Gaussian elimination or finding the inverse of a matrix can be more efficient.

For this particular system, we will primarily focus on the substitution and elimination methods due to their straightforward application.

Solving the System Using the Elimination Method

The elimination method is particularly well-suited for this system due to the coefficients of the variables. Observe that the coefficients of x in the two equations are 2 and -5, and the coefficients of y are 8 and -20. Notice a relationship? The second equation's coefficients are multiples of the first equation's coefficients. This suggests that elimination can be achieved with a simple manipulation.

Let's multiply the first equation by 5 and the second equation by 2. This will make the coefficients of x opposites of each other:

• First equation multiplied by 5: 5 * (2x + 8y) = 5 * 6 which simplifies to 10x + 40y = 30
• Second equation multiplied by 2: 2 * (-5x - 20y) = 2 * (-15) which simplifies to -10x - 40y = -30

Now, add the two modified equations together:

(10x + 40y) + (-10x - 40y) = 30 + (-30)

This simplifies to:

0 = 0

This result is a significant indicator. The equation 0 = 0 is always true, regardless of the values of x and y. This means that the two original equations are not independent; they are essentially representing the same line. The system has infinitely many solutions.

Understanding Dependent Systems

When a system of linear equations results in an identity like 0 = 0, it signifies a dependent system. A dependent system occurs when the equations represent the same line or multiples of each other. In graphical terms, the two lines would overlap completely. This leads to an infinite number of solutions because every point on the line satisfies both equations.

In contrast, an independent system has a unique solution, represented by the single point where the lines intersect. An inconsistent system has no solutions, meaning the lines are parallel and never intersect.

Solving the System Using the Substitution Method

To further illustrate the nature of this system, let's apply the substitution method. We'll start by solving the first equation for x:

2x + 8y = 6
2x = 6 - 8y
x = 3 - 4y

Now, substitute this expression for x into the second equation:

-5x - 20y = -15
-5(3 - 4y) - 20y = -15

Distribute the -5:

-15 + 20y - 20y = -15

Simplify the equation:

-15 = -15

Again, we arrive at an identity, -15 = -15, which is always true. This confirms our previous finding that the system is dependent and has infinitely many solutions. The substitution method further solidifies the conclusion that the equations are essentially representing the same line.

Expressing the Solution Set

Since the system has infinitely many solutions, we need a way to express the solution set. We can do this by expressing one variable in terms of the other. We already solved the first equation for x in terms of y: x = 3 - 4y. This provides us with a general form for the solutions.

The solution set can be expressed as ordered pairs (x, y) where x is given by 3 - 4y and y can be any real number. Mathematically, we can write this as:

{(x, y) | x = 3 - 4y, y ∈ ℝ}

This notation means