Problem 70 - Precalculus

Question

Solve the system of equations by elimination. $$ \begin{array}{l} 4 x-y=12 \\ 4 x-y=24 \end{array} $$ Solution: Multiply Equation ( 1 ) by -1 Add the result to Equation (2) $$\begin{array}{rr} -4 x+y= & -12 \\ -4 x+y= & -12 \\ 4 x-y= & 24 \\ \hline 0= & 12 \end{array}$$ Answer: Infinitely many solutions. This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The system has no solutions because the equations are inconsistent.

Step 1: Identify the Equations

We have the system of equations: \( 4x - y = 12 \) and \( 4x - y = 24 \). These equations are written in the form \( Ax + By = C \).

Step 2: Compare the Equations

Look at the coefficients of \(x\) and \(y\). Both equations have the same coefficients: 4 for \(x\) and -1 for \(y\). However, the constant terms (12 and 24) are different.

Step 3: Analyze the System

Since the left-hand sides of the equations are identical (i.e., they have the same coefficients for \(x\) and \(y\)), but the right-hand sides (the constants) are different, the system is inconsistent. This means it has no solution.

Step 4: Recognize the Error

The error in the original solution was asserting that the system has infinitely many solutions. In reality, the system of equations has no possible values for \(x\) and \(y\) that satisfy both equations simultaneously because the left-hand sides are identical but yield different results (12 and 24).

Key Concepts

Elimination Method

The elimination method is one of the primary techniques used to solve systems of equations. It's particularly useful when you want to eliminate one of the variables so you can solve for the other. Here’s a breakdown of how it works:

• Align the Equations: Start by writing the system of equations in a standard form, usually as \(Ax + By = C\). This helps in comparing the coefficients of the variables.
• Make the Coefficients Opposite: Manipulate the equations so that you can eliminate one variable. This often involves multiplying one or both equations by a specific number that makes the coefficients of one variable equal and opposite.
• Add or Subtract the Equations: Use addition or subtraction to cancel one of the variables out. This results in a single equation with one variable, which can then be solved to find a particular value.

In the given exercise, the attempt was to eliminate variable \(y\) by making its coefficients identical and opposite, then add them to cancel \(y\). However, since both equations were initially identical, the elimination revealed an issue: inconsistency, not a solution.

Inconsistent System

An inconsistent system of equations occurs when there are no simultaneous solutions for the variables that can satisfy both equations. This can happen due to contradictory information presented by the equations.

• Identical Coefficients, Different Constants: If two equations have the same coefficients for the variables, but different constants on the right side, this indicates inconsistency. It means that the equations describe parallel lines that will never intersect, hence no solution exists.
• Graphical Interpretation: On a graph, an inconsistent system generally translates to parallel lines that have no point in common.

In the exercise provided, the left-hand sides of the equations, \(4x - y\) were identical, while the constants differed (12 and 24). This means that no value of \(x\) and \(y\) could work for both equations simultaneously, thus making the system inconsistent.

Solution Analysis

When analyzing a system of equations, recognizing the type of solution is crucial. There are three potential outcomes for a system: exactly one solution, no solution, or infinitely many solutions.

• Exactly One Solution: This occurs when the system of equations represents two lines that intersect at a single point. The coefficients and constants differ, supporting a unique intersection point.
• No Solution: Known as an inconsistent system, this happens when the equations describe parallel lines that never meet. Our exercise falls under this category, where identical left-hand sides but differing constants point to parallel lines.
• Infinitely Many Solutions: This happens when the equations are essentially the same, i.e., multiples of each other, leading to coinciding lines that overlap completely.

The error in the provided solution was in concluding that the system had infinitely many solutions, whereas the differing constants should have indicated no solution. This shows the importance of careful comparison of coefficients and constants during solution analysis.