Problem 95 - Algebra and Trigonometry Real Mathematics, Real People

Question

Create systems of two linear equations in two variables that have (a) no solution, (b) one distinct solution, and (c) infinitely many solutions.

Step-by-Step Solution

Answer

An example of systems with (a) no solution: \( y = 2x + 3 \), \( y = 2x - 1 \); (b) one distinct solution: \( y = 2x + 3 \), \( y = -x + 1 \); and (c) infinitely many solutions: \( y = 2x + 3 \), \( y = 2(x + 1.5) - 3 \).

Step 1: Creating a System with No Solution

Systems of equations that have no solutions are called inconsistent systems. To create such a system, it's needed to make sure the two lines are parallel; meaning they have the same slope but different y-intercepts. An example of this would be \( y = 2x + 3 \) and \( y = 2x - 1 \). These two lines will never intersect because they are parallel.

Step 2: Creating a System with One Distinct Solution

To create a system with one distinct solution, the two lines should intersect at exactly one point. This means that they should have different slopes. They can have the same or different y-intercepts. An example of such a system would be \( y = 2x + 3 \) and \( y = -x + 1 \). They will intersect at one point, thus having one unique solution.

Step 3: Creating a System with Infinitely Many Solutions

For a system to have infinitely many solutions, the two lines should coincide, meaning they are actually the same line represented in different ways. This means they have the same slope and the same y-intercept. An example would be \( y = 2x + 3 \) and \( y = 2(x + 1.5) - 3 \). If the equations are simplified, it shows that both are the same equation, hence they will have infinitely many solutions.

Key Concepts

Inconsistent Systems

Inconsistent systems are a fascinating aspect of linear algebra. They refer to systems of linear equations that do not have any solutions. This occurs when the lines represented by the equations are parallel. Parallel lines have identical slopes, but different y-intercepts, which is why they never meet. It's like two train tracks that run side by side endlessly without ever intersecting.

To design an inconsistent system, you could take two equations like these:

• Equation 1: \( y = 2x + 3 \)
• Equation 2: \( y = 2x - 1 \)

Both equations have the same slope of 2, aligning them in parallel, yet they differ in their y-intercepts. This distinction in intercept values ensures that the lines will never cross. Students should practice identifying these characteristics to grasp the concept of inconsistent systems fully.

Distinct Solution

A distinct or unique solution in a system of equations occurs when two lines intersect at exactly one point. This happens when the lines have different slopes. Since slope represents the angle of the line, two different slopes ensure that the lines will cross at one point and thus have one distinct solution.

To create a system with a distinct solution, try varying the slope of each line:

• First Equation: \( y = 2x + 3 \)
• Second Equation: \( y = -x + 1 \)

Here, the slopes differ (2 and -1), ensuring the lines will intersect at one point. This intersection is the unique solution where both equations are satisfied. Practicing with different slope values will help visualize how these intersections occur at distinct points on a graph.

Infinitely Many Solutions

A system with infinitely many solutions is one where the equations represent the same line. Thus, any point on the line is a solution. This means they have not only identical slopes but also matching y-intercepts, effectively making them coincident lines.

Such systems can be written differently, yet convey the same line:

• First Equation: \( y = 2x + 3 \)
• Second Equation: \( y = 2(x + 1.5) - 3 \)

By simplifying the second equation, you would find it is indeed the same as the first. This demonstrates that every point on these lines is a shared solution, leading to infinitely many solutions. Understanding this concept is crucial for recognizing consistent and dependent systems, which cover the entirety of the line they form. Exploring different representations will solidify this understanding in students.