Solving Dependent Systems Of Equations Expressing Solutions

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In mathematics, solving systems of linear equations is a fundamental skill with applications across various fields, from engineering and physics to economics and computer science. A system of linear equations consists of two or more linear equations involving the same variables. The solution to a system of linear equations is the set of values for the variables that satisfy all equations simultaneously. In this article, we will delve into the process of solving a specific system of equations, explore the concept of dependent systems, and learn how to express the solution in such cases.

Understanding Systems of Linear Equations

Before we dive into solving the system, let's establish a solid understanding of what systems of linear equations are. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variables in a linear equation can only have a power of 1, and there are no products or other functions of the variables. A system of linear equations is a collection of two or more linear equations that share the same variables. The goal is to find the values for these variables that make all equations in the system true at the same time.

Methods for Solving Systems of Linear Equations

There are several methods for solving systems of linear equations, each with its own advantages and disadvantages. The most common methods include:

  • Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can be easily solved. Once the value of one variable is found, it can be substituted back into either of the original equations to find the value of the other variable.
  • Elimination Method: This method involves manipulating the equations in the system so that the coefficients of one of the variables are opposites. When the equations are added together, this variable is eliminated, leaving a single equation with one variable. This equation can then be solved, and the value of the variable can be substituted back into either of the original equations to find the value of the other variable.
  • Graphical Method: This method involves graphing each equation in the system on the same coordinate plane. The solution to the system is the point of intersection of the lines. This method is particularly useful for visualizing the solutions of systems of two equations with two variables.
  • Matrix Methods: For larger systems of equations, matrix methods such as Gaussian elimination or matrix inversion are often used. These methods provide a systematic way to solve systems of equations using matrix operations.

Solving the System of Equations: A Detailed Walkthrough

Now, let's tackle the specific system of equations presented:

\left\{
\begin{array}{l}
2 x+3 y=3 \\
6 x+9 y=9
\end{array}
\right.

This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations simultaneously. We will demonstrate the solution using the elimination method.

Step 1: Analyze the Equations

Observe the coefficients of the variables in both equations. Notice that the coefficients of x in the second equation (6) are three times the coefficients of x in the first equation (2). Similarly, the coefficients of y in the second equation (9) are three times the coefficients of y in the first equation (3). This suggests a potential relationship between the two equations.

Step 2: Manipulate the Equations

To apply the elimination method, we need to make the coefficients of one of the variables opposites. Let's focus on eliminating x. To do this, we can multiply the first equation by -3:

-3 * (2x + 3y) = -3 * 3

This simplifies to:

-6x - 9y = -9

Now we have the following system:

\left\{
\begin{array}{l}
-6 x-9 y=-9 \\
6 x+9 y=9
\end{array}
\right.

Step 3: Eliminate a Variable

Add the modified first equation to the second equation:

(-6x - 9y) + (6x + 9y) = -9 + 9

This simplifies to:

0 = 0

Step 4: Interpret the Result

The result 0 = 0 is a true statement, but it doesn't give us specific values for x and y. This indicates that the system is dependent. A dependent system has infinitely many solutions. This occurs when the equations in the system represent the same line or are multiples of each other.

Dependent Systems and Expressing the Solution

When a system of equations is dependent, it means that the equations are essentially the same. In our case, if you divide the second original equation (6x + 9y = 9) by 3, you get the first equation (2x + 3y = 3). This confirms that the equations are multiples of each other and represent the same line.

Expressing the Solution for a Dependent System

Since there are infinitely many solutions, we cannot provide specific numerical values for x and y. Instead, we express the solution in terms of a parameter. This involves solving one of the equations for one variable in terms of the other. Let's solve the first equation (2x + 3y = 3) for x:

2x = 3 - 3y

Divide both sides by 2:

x = (3 - 3y) / 2

Now, we can express the solution set as follows:

\{(x, y) | x = (3 - 3y) / 2\}

This notation represents the set of all ordered pairs (x, y) such that x is equal to (3 - 3y) / 2. In simpler terms, for any value we choose for y, we can find a corresponding value for x that satisfies both equations. This confirms the infinite number of solutions in a dependent system.

Alternative Representation

We could also express the solution by solving for y in terms of x. Let's do that:

Starting with 2x + 3y = 3, subtract 2x from both sides:

3y = 3 - 2x

Divide both sides by 3:

y = (3 - 2x) / 3

Now, we can express the solution set as:

\{(x, y) | y = (3 - 2x) / 3\}

Both representations are valid and convey the same information: the system has infinitely many solutions, and the relationship between x and y is defined by the equation we used to solve for one variable in terms of the other.

Key Takeaways

  • A system of linear equations can have one solution, no solution (inconsistent system), or infinitely many solutions (dependent system).
  • A dependent system occurs when the equations are multiples of each other, representing the same line.
  • The solution to a dependent system is expressed in terms of a parameter, showing the relationship between the variables.
  • The elimination method is a powerful technique for solving systems of linear equations, especially when dealing with dependent systems.

Conclusion: Mastering Systems of Linear Equations

Solving systems of linear equations is a crucial skill in mathematics with wide-ranging applications. Understanding the different types of systems, such as dependent systems, and knowing how to express their solutions is essential. By mastering techniques like the elimination method and understanding the concept of parameterization, you can confidently tackle a wide range of problems involving systems of linear equations. Remember, practice is key to honing your skills and developing a deeper understanding of these concepts. So, keep exploring, keep practicing, and keep solving!

Solving Systems of Equations: Determining Dependence and Expressing Solutions

Let's dive into the world of systems of equations! Solving systems of equations is a fundamental concept in algebra, and understanding the nature of solutions – whether they are unique, nonexistent, or infinite – is crucial. In this article, we will explore a specific system of equations and demonstrate how to determine if it is dependent, and if so, how to express its solution in a meaningful way. This exploration will enhance your problem-solving toolkit and deepen your understanding of linear algebra concepts.

Understanding the Basics of Systems of Equations

At its core, a system of equations is a set of two or more equations containing the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. Geometrically, each equation in a system represents a line (in two dimensions), a plane (in three dimensions), or a hyperplane (in higher dimensions). The solution to the system corresponds to the point(s) where these geometric objects intersect. The number of solutions a system can have can vary greatly depending on the nature of the equations and their relationships with each other.

There are primarily three possibilities for the number of solutions a system of linear equations can possess:

  1. Unique Solution: The system has exactly one solution, meaning there is only one set of values for the variables that satisfies all equations. Geometrically, this corresponds to the lines (or planes, etc.) intersecting at a single point.
  2. No Solution (Inconsistent System): The system has no solution, indicating that there is no set of values for the variables that can satisfy all equations simultaneously. Geometrically, this corresponds to the lines (or planes, etc.) being parallel and never intersecting.
  3. Infinitely Many Solutions (Dependent System): The system has infinitely many solutions, which means there are countless sets of values for the variables that satisfy all equations. Geometrically, this corresponds to the lines (or planes, etc.) coinciding, meaning they are essentially the same line or plane.

Methods for Solving Systems of Equations

Several methods are available for solving systems of equations, each with its strengths and weaknesses. The most common methods include:

  • Substitution: This method involves solving one equation for one variable and then substituting that expression into another equation. This reduces the system to a smaller set of equations with fewer variables.
  • Elimination: Also known as the addition or subtraction method, this technique involves manipulating the equations in the system to eliminate one of the variables. This is achieved by multiplying one or both equations by constants so that the coefficients of one variable are additive inverses, and then adding the equations together.
  • Graphical Methods: For systems with two variables, graphing the equations can provide a visual representation of the solutions. The intersection points of the graphs represent the solutions to the system.
  • Matrix Methods: For larger systems, matrix methods like Gaussian elimination or matrix inversion offer a systematic and efficient approach to finding solutions.

Analyzing the Given System of Equations

Now, let's turn our attention to the system of equations we want to analyze:

\left\{
\begin{array}{l}
2 x+3 y=3 \\
6 x+9 y=9
\end{array}
\right.

This system consists of two linear equations with two variables, x and y. Our goal is to determine if this system is dependent and, if so, express its solution. Let's use the elimination method to analyze the system.

Applying the Elimination Method

  1. Observe the Coefficients: Notice that the coefficients of x in the second equation (6) are three times the coefficients of x in the first equation (2). The same relationship holds for the coefficients of y (9 is three times 3). This observation is a strong indicator that the system might be dependent.

  2. Manipulate the Equations: To apply the elimination method effectively, we need to make the coefficients of one of the variables opposites. Let's choose to eliminate x. Multiply the first equation by -3:

    -3 * (2x + 3y) = -3 * 3

    This simplifies to:

    -6x - 9y = -9

    Now our system looks like this:

    \left\{
\begin{array}{l}
-6 x-9 y=-9 \\
6 x+9 y=9
\end{array}
\right.

  3. Eliminate a Variable: Add the modified first equation to the second equation:

    (-6x - 9y) + (6x + 9y) = -9 + 9

    This simplifies to:

    0 = 0

Interpreting the Result: Dependence

The result 0 = 0 is a true statement, but it doesn't give us specific values for x and y. This is the hallmark of a dependent system. When you eliminate a variable and end up with an identity (a true statement that doesn't involve variables), it means the equations in the system are not independent. In this case, the two equations represent the same line. Dividing the second equation (6x + 9y = 9) by 3 yields the first equation (2x + 3y = 3), further confirming their dependence.

Expressing the Solution for a Dependent System

Since a dependent system has infinitely many solutions, we cannot provide a single pair of values for x and y. Instead, we express the solution in terms of a parameter. This involves solving one of the equations for one variable in terms of the other. Let's solve the first equation (2x + 3y = 3) for x:

  1. Subtract 3y from both sides:

    2x = 3 - 3y

  2. Divide both sides by 2:

    x = (3 - 3y) / 2

Now we have x expressed in terms of y. We can represent the solution set as:

\{(x, y) | x = (3 - 3y) / 2\}

This notation reads: