Problem 80 - College Algebra with Modeling and Visualization

Question

Use elimination to solve the system of equations, if possible. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. Support your results graphically or numerically. $$ \begin{array}{c} x-3 y=1 \\ 2 x-6 y=2 \end{array} $$

Step-by-Step Solution

Answer

The system is consistent and dependent.

Step 1: Write the System of Equations

We have the following system of equations to solve:1. \( x - 3y = 1 \)2. \( 2x - 6y = 2 \)

Step 2: Eliminate one variable

We aim to eliminate one of the variables by applying the elimination method. Let's subtract equation 1 from equation 2 to eliminate variable \( x \). Equation 2: \( 2x - 6y = 2 \)2 times Equation 1: \( 2(x - 3y) = 2(1) \) gives: \( 2x - 6y = 2 \). Subtracting these equations yields:\((2x - 6y) - (2x - 6y) = 2 - 2 \)\( 0 = 0 \)

Step 3: Interpretation of the Result

The result \( 0 = 0 \) is a true statement and indicates that the two equations are essentially the same (or multiples of each other). This implies that the system of equations is consistent and dependent.

Step 4: Numerical Substantiation

To numerically verify, you can solve one equation for \( x \):\( x = 3y + 1 \), then substitute back into one of the original equations to verify consistency. Since we derive a true statement (\( 0 = 0 \)), both equations represent the same line and are consistent and dependent.

Step 5: Graphical Representation (Optional)

Graph both lines to see that they overlap completely, confirming they represent the same line in coordinate geometry. Every solution of one equation is also a solution for the other, further confirming dependency.

Key Concepts

System of Equations

A system of equations is a set of two or more equations with the same variables. Solving a system means finding the values for the variables that make all the equations true at the same time. There are several ways to solve systems of equations, such as substitution, elimination, and graphing. In the elimination method, we manipulate the equations to cancel out one of the variables, making it easier to solve for the remaining variable. For instance, in the given system:

• 1. \( x - 3y = 1 \)
• 2. \( 2x - 6y = 2 \)

We multiplied Equation 1 by 2 and subtracted it from Equation 2, which helped us eliminate the variable \( x \). This approach simplifies the system to a form that allows us to determine relationships between the equations. Systems of equations are foundational in algebra as they model real-world problems that require simultaneous solutions.

Consistent and Dependent Systems

In terms of systems of equations, consistency and dependency are important concepts to understand. A consistent system is one that has at least one solution. It is also possible for consistent systems to have exactly one solution, making them consistent and independent. However, in our exercise, we find something different: the system exhibits characteristics of a dependent system.

Dependent systems result when the equations are essentially the same, just presented differently or as multiples of one another. The solution is that there are infinitely many solutions, as the equations describe the same line.

For the system here, the elimination process led to the statement \( 0 = 0 \), which is true and indicates infinitely many solutions pointing to dependency. Therefore, understanding whether a system of equations is consistent and dependent helps predict how many solutions exist and whether the equations intersect as independent lines or overlap as one.

Graphical Representation

Graphical representation is a visual tool that helps understand the relationship between equations in a system. When you graph the equations from the exercise:

• Equation 1: \( x - 3y = 1 \)
• Equation 2: \( 2x - 6y = 2 \)

You'll find that they are actually the same line. The lines coincide perfectly, meaning that every solution of one equation is also valid for the other.

Graphically, this overlapping shows the dependent nature of the system as both equations describe an identical line. If the lines intersected at a single point they would be consistent and independent, with that point being the sole solution. Graphing provides a clear and intuitive way to visualize how equations interact, making it an essential part of solving and understanding systems of equations in both classroom settings and practical applications.