Problem 74 - 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. $$ \begin{array}{r} 4 x+2 y=10 \\ -2 x-y=10 \end{array} $$

Step-by-Step Solution

Answer

The system is inconsistent; no solution exists.

Step 1: Align the Equations for Elimination

We have the system of equations: \( 4x + 2y = 10 \) and \( -2x - y = 10 \). To apply elimination, notice that the second equation is already aligned to cancel terms when we add it to the first equation.

Step 2: Add the Equations to Eliminate a Variable

Add the two equations together: \( (4x + 2y) + (-2x - y) = 10 + 10 \). This simplifies to \( 2x + y = 20 \).

Step 3: Simplify the Resulting Equation

The result of the addition is \( 2x + y = 20 \). There are no variables to solve immediately, and the key is to compare this new equation to the original system.

Step 4: Try Substituting to Find a Solution

Substitute \( y = 20 - 2x \) from the new equation into one of the original equations (like \( 4x + 2y = 10 \)) to see if a solution exists. Substituting that gives: \( 4x + 2(20 - 2x) = 10 \)which simplifies to \( 4x + 40 - 4x = 10 \), further simplified to \( 40 = 10 \).

Step 5: Analyze the Results for Consistency

Since \( 40 = 10 \) is false, the system is inconsistent, indicating there are no solutions that satisfy both equations.

Key Concepts

System of Equations

A system of equations consists of two or more equations that share the same set of variables. In such systems, we are interested in finding a common solution to all the equations involved. There are several ways to solve these systems, such as graphically, using substitution, or the elimination method. In the elimination method, you aim to remove one of the variables by adding or subtracting equations, neatly simplifying the problem.

In our system, we are dealing with two linear equations:

• Equation 1: \( 4x + 2y = 10 \)
• Equation 2: \( -2x - y = 10 \)

These represent straight lines, and solving the system means finding the point(s) where the lines intersect. If they intersect at a single point, there is a unique solution. If the lines are parallel, then there is no solution because they never meet. Lastly, if the lines are overlapping or the same, there are infinitely many solutions because every point is a solution.

Inconsistent Systems

An inconsistent system is a system of equations that has no solution because the equations contradict each other. This means there is no pair of \((x, y)\) values that will satisfy both equations simultaneously. We typically identify an inconsistent system when the process of solving leads to a false statement like "40 = 10," which is clearly invalid.

In this exercise, we added the equations: \[ (4x + 2y) + (-2x - y) = 10 + 10 \2x + y = 20 \]Substituting \( y = 20 - 2x \) into \( 4x + 2y = 10 \) resulted in the equation \( 40 = 10 \), showing no valid solutions exist. Graphically, such a conclusion is represented by parallel lines that never meet.

• Equation result: A contradiction signifies no common solution.
• Graphical result: Lines do not intersect.

When systems display this behavior, they are called inconsistent.

Dependent and Independent Equations

In system discussions, the terms dependent and independent help us understand the relationship between equations. This determines whether we have a unique solution, no solution, or infinite solutions.

• **Independent Equations:** These equations intersect at a single point. A solution exists, and it is unique.
• **Dependent Equations:** These essentially represent the same line or are multiples of one another, meaning every solution of one equation is also a solution of the other, resulting in infinitely many solutions.

In our specific problem, the lack of a valid intersection led us to classify the system as inconsistent. Analyzing equations like ours is crucial because it helps us confirm the relationship between equations and the potential existence of solutions.
Knowing the difference between dependent, independent, and inconsistent systems is key when tackling algebraic problems, as it shapes the strategy for approaching and solving them effectively.