Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Explain how one can determine whether a system is inconsistent or has dependent equations when using the substitution or elimination method.

Step-by-Step Solution

Answer

Use substitution or elimination: A contradiction means the system is inconsistent; a tautology means it's dependent.

Step 1: Understand the Terminology

In a system of equations, if there are no solutions, the system is called inconsistent. If all equations are essentially the same (i.e., every equation is a scalar multiple of another), the system has dependent equations and infinitely many solutions.

Step 2: Use the Substitution Method

Solve one of the equations for one variable and substitute it into the other equation. Simplify to see if the variables can be solved. If the equations simplify to a contradiction, such as 0 = 5, the system is inconsistent.

Step 3: Use the Elimination Method

Multiply one or both equations by appropriate constants to eliminate one variable by addition or subtraction of equations. Simplify what's left. If the remaining equation is a contradiction, the system is inconsistent. If the remaining equation simplifies to a tautology, like 0 = 0, then the system is dependent and has infinitely many solutions.

Step 4: Recognize the Signs of Inconsistency and Dependence

When using either method, if you arrive at a false statement after all variables are eliminated (e.g., 0 = 5), the system is inconsistent. If you arrive at a true yet trivial relation (e.g., 0 = 0), all original equations gave the same line, indicating dependent equations.

Key Concepts

Inconsistent Systems

In the realm of algebra, a system of equations is deemed inconsistent if it has no solutions. This typically occurs when the lines represented by the equations never intersect. You can visualize this as two parallel lines that never meet. For example, the equations might simplify to something impossible like `0 = 5`. This straightforward contradiction tells us that these equations cannot be solved together, marking the system as inconsistent. When using either the substitution or elimination method, arriving at such a contradiction means the system is inconsistent.

Dependent Equations

Dependent equations in a system refer to cases where all the equations essentially describe the same geometric entity, like a single line in a 2D plane. When all equations are multiples of each other, they don't provide any new information. As a result, the system has infinitely many solutions, existing along the line described. In practice, after applying the substitution or elimination methods, a system will show its dependency if it simplifies to a true but trivial statement, such as `0 = 0`. This outcome means that one equation can be derived from the others, indicating they are dependent.

Substitution Method

The substitution method is a strategic way to find solutions in systems of equations. Here, you solve one equation for a single variable and then replace that variable in the other equation. This method is particularly useful in cases where one variable is already isolated in one of the equations. The strength of this approach lies in its simplicity and ability to easily detect inconsistencies or dependencies:

• If, after substitution, you find an absurdity, like `0 = 5`, the system is inconsistent.
• Conversely, if everything cancels out leading to a true statement like `0 = 0`, you're dealing with dependent equations.

Elimination Method

The elimination method is an efficient way to solve a system of equations by adding or subtracting equations to eliminate a variable. It often requires multiplying one or both of the equations by specific numbers to line up coefficients. This method shines in providing a clear path to the solution:

• If the operation results in a false formula, such as `0 = 3`, it indicates an inconsistent system, showing that the equations have no intersection point.
• If the variables cancel leaving a universally true equation like `0 = 0`, then the equations are dependent, meaning each equation represents the same line.