Problem 11 - Elementary Algebra

Question

Use the elimination-by-addition method to solve each system. $$\left(\begin{array}{l}7 x-2 y=4 \\ 7 x-2 y=9\end{array}\right)$$

Step-by-Step Solution

Answer

No solution exists; the system is inconsistent.

Step 1: Analyze the system of equations

We have two equations in the system: \(7x - 2y = 4\) and \(7x - 2y = 9\). Notice that both equations have the same left-hand side but different right-hand sides.

Step 2: Identify inconsistency

Because both equations have the same left-hand side, subtraction will eliminate both terms, resulting in \(0 = 5\) (\(9 - 4 = 5\)). This is a contradiction, indicating there's no solution to this system.

Key Concepts

Elimination Method

The elimination method is a common technique used to solve systems of linear equations. It involves eliminating one of the variables by adding or subtracting the equations from each other. This simplifies the system to a single variable equation, making it easier to solve. Here's a basic rundown of the method:

• Align the equations, ensuring variables are in the same columns.
• Decide which variable to eliminate. Oftentimes, it’s easier to target the variable with the same or opposite coefficients in both equations.
• Add or subtract the equations. Multiplying one or both equations by a constant might be necessary to get the coefficients to align.
• Solve the resulting single-variable equation.
• Substitute the solution back into one of the original equations to find the second variable.

In our example, the equations were already aligned with identical coefficients for both terms. However, subtraction indicated inconsistency, which we'll explore next.

Inconsistent System

An inconsistent system occurs when there is no solution possible for the given set of equations. This contradiction arises when the elimination method leads to a false statement, such as 0 = 5. This is a clear signal that the system is inconsistent. Several scenarios can hint at inconsistency:

• Parallel lines upon graphing, which never intersect, and thus, have no solution.
• Equations with the same left-hand side but different right-hand sides, which imply they cannot hold true simultaneously.

In your problem, both equations shared identical terms on one side but differed in the results. When the system is inconsistent, it generally indicates that the lines are parallel, reinforcing that there are no shared solutions.

Algebra Problem Solving

Algebra problem-solving involves several methods like the elimination method, substitution, and graphing. These techniques help unravel the values of variables from equations and systems of equations. For any algebra problem, keep these strategies in mind:

• Clearly identify what is known and unknown in the problem.
• Choose the best method for solving, whether it is elimination, substitution, or another technique.
• Work systematically, line by line, to avoid common mistakes.
• Check your solution by plugging values back into the original equations to ensure they satisfy all given conditions.

Even when a system like the one provided is inconsistent, it's essential to reach this conclusion mathematically to understand why there is no solution, helping to reinforce mathematical logic and reasoning skills.