Problem 21 - Precalculus: Mathematical for Calculus

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{array}{r} x+4 y=8 \\ 3 x+12 y=2 \end{array}\right.$$

Step-by-Step Solution

Answer

The system has no solution.

Step 1: Analyze the System of Equations

First, look at the given system of equations:\ 1) \(x + 4y = 8\)\ 2) \(3x + 12y = 2\)\ Notice that the second equation is a multiple of the first equation, except for the constant term on the right side.

Step 2: Compare the Equations for Dependency

In a dependent system, multiplying one equation by a constant should yield the other equation. Try to manipulate the first equation by multiplying it:Multiply the first equation by 3: \((x + 4y = 8) imes 3\) which gives \(3x + 12y = 24\)Compare this to the second equation \(3x + 12y = 2\). The left sides match, but the right sides do not (24 vs 2), indicating these equations are inconsistent.

Step 3: Determine System Consistency

Since the equations \(3x + 12y = 24\) and \(3x + 12y = 2\) represent parallel lines (same slope, different intercepts), these equations have no points of intersection.The system is inconsistent and has no solution.

Key Concepts

Inconsistent Systems

Inconsistent systems in linear algebra are systems of equations that have no common solution. This means there is no set of values that satisfy all the equations simultaneously. This usually happens when the equations represent parallel lines.
When two lines in the Cartesian plane are parallel, they have the same slope but different intercepts, leading them never to meet at any point. For example, consider the equations:

• \(x + 4y = 8\)
• \(3x + 12y = 2\)

The second equation is a scaled version of the first one but with a different constant term. When equations have this relationship, they are inconsistent because they describe parallel lines that do not intersect. In this example, multiplying the first equation by 3 results in the equation \(3x + 12y = 24\), which corresponds to a different line from \(3x + 12y = 2\). Hence, they are inconsistent, with no solutions existing to satisfy both equations simultaneously.

Dependent Equations

Dependent equations arise in systems of equations where one equation can be converted into another by multiplying by a constant, leading to infinitely many solutions. In essence, they represent the same line with the same slope and intercept. However, it is crucial to note that dependent equations share equivalent expressions and do not belong to the inconsistent system category.
Consider, however, a different scenario than in the exercise. If manipulating the equation \(x + 4y = 8\) provided \(3x + 12y = 24\) under a correct transformation rather than \(3x + 12y = 2\), the two equations would describe the same geometric line on a graph.

• This indicates they are entirely dependent on one another.
• Every point on the line \(x + 4y = 8\) is also on the line \(3x + 12y = 24\), hence an infinite number of solutions.

This scenario is not applicable to the currently problematic system because altering and comparing the equations led to the discovery of their inconsistencies rather than dependencies.

Parallel Lines

Parallel lines are lines in the same plane that never intersect. They have identical slopes but different y-intercepts. In a system of equations that is inconsistent, the corresponding lines describe parallel lines in the graph.
Consider the example given in the exercise: the equations \(3x + 12y = 24\) and \(3x + 12y = 2\) both describe lines with the same slope, indicating their parallel nature.

• This means they remain equidistant at all points.
• Because they don't intersect, the system has no solutions, establishing it as inconsistent.
• Parallel lines reveal a direct visual interpretation of how and why inconsistent systems do not yield solutions.

Understanding parallel lines is vital since it highlights the geometric representation of inconsistent systems, making it easier to recognize when equations can never have a common solution.