Problem 72 - 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. $$ \begin{array}{r} 2 x+y=15 \\ x-y=0 \end{array} $$

Step-by-Step Solution

Answer

The system is consistent and independent, with a solution \((x, y) = (5, 5)\).

Step 1: Align the System of Equations

First, let's write down the system of equations clearly:\[ \begin{align*} 2x + y &= 15 \ x - y &= 0 \end{align*} \] Our goal is to use elimination to solve this system.

Step 2: Eliminate One Variable

To eliminate one variable, we can add the two equations: The second equation can be rewritten as:\[ x = y \]Now, add this equation to the first equation:\[ \begin{align*} 2x + y &= 15 \ x - y &= 0 \end{align*} \]Adding these equations gives us:\[ 3x = 15 \]

Step 3: Solve for One Variable

Divide both sides of the equation obtained from elimination by 3:\[ x = \frac{15}{3} = 5 \]Thus, we have found the value of \(x\).

Step 4: Substitute Back to Find the Other Variable

Substitute \(x = 5\) back into the equation \(x = y\):\[ y = 5 \]Thus, both \(x\) and \(y\) have been found to be 5.

Step 5: Analyze the Consistency and Dependence

The solution \((x, y) = (5, 5)\) satisfies both the original equations, indicating the system is consistent. Since the two equations are not multiples of each other, they are independent.

Key Concepts

Elimination Method

The elimination method is one of the key techniques used to solve systems of linear equations. It focuses on removing one variable so you can easily solve for the other. Here, we aim to eliminate one variable by adding or subtracting equations. In our given system: \[ 2x + y = 15 \]\[ x - y = 0 \]we started by aligning the equations and then adding them to cancel out the \( y \) variable. By rewriting the second equation as \( x = y \), and adding it to the first, we achieved: \[ 3x = 15 \]allowing us to solve for \( x \). Remember:

• Elimination simplifies a system into fewer variables.
• This method is particularly useful when variables have the same or opposite coefficients.

Once you've solved for one variable, substitution helps solve for the other.

Consistent System

When we discuss a consistent system of equations, we refer to a scenario where there is at least one set of values that satisfy all equations in the system. In our example, the solution \((x, y) = (5, 5)\) satisfies both: \[ 2x + y = 15 \]\[ x - y = 0 \]Such a result indicates consistency. There are different types of consistent systems:

• Unique solution: Only one set of solutions, as in this case.
• Infinitely many solutions: Many solutions satisfy the system, usually when the equations are dependent.

This means our system here not only has solutions but specifically a unique solution, solidifying its consistency.

Independent Equations

In a system of equations, independence implies that no equation can be written as a multiple of another. Independent systems tend to have a unique solution. In our system:\[ 2x + y = 15 \]\[ x - y = 0 \]none of the equations is a scalar multiple of the other. This feature ensures that the lines represented by these equations intersect at precisely one point, yielding a unique solution, \((5, 5)\).

• Independent equations guarantee distinct solutions.
• The method of solving might change, but independence provides a consistent foundational structure.

Analyzing the coefficients and constants can easily confirm independence.

Graphical Representation

Visualizing systems of equations graphically can provide an immediate sense of their solutions. The graph of a system like ours, given by:- \( 2x + y = 15 \)- \( x - y = 0 \)results in two lines on a coordinate plane. The point at which they intersect is the solution to the system. In this case, they intersect at the point \((5, 5)\).A few benefits of graphical representation:

• Visual clarity: Instantly see whether lines intersect, are parallel, or coincide.
• Immediate insight into whether the system is consistent or inconsistent.

Using graphs, even a quick sketch, reaffirms the values found algebraically and strengthens understanding.