Problem 41 - Introductory Algebra for College Students

Question

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}y=0 \\ y=4\end{array}\right.$$

Step-by-Step Solution

Answer

The solution is ∅ (empty set)

Step 1: Graph the first line

The first line y=0 is simply the x-axis. The line crosses the x-axis at all points where y=0.

Step 2: Graph the second line

The second line y=4, is a horizontal line that crosses the y-axis at the point (0,4). The line is at a constant height of 4 on the y-axis.

Step 3: Observe the Intersection

There is no point where these two lines intersect, because they are two parallel horizontal lines.

Step 4: Express the Solution in Set Notation

Since the lines do not intersect, there is no common solution to the two equations and the system is inconsistent. In set notation, this is expressed as ∅ (the empty set, meaning there's no solution).

Key Concepts

Graphing Method

When solving a system of equations using the graphing method, we visually represent each equation on a coordinate plane. This helps us understand the relationship between the equations by observing where the lines intersect, if they do at all. Here's a simple breakdown of the process:

• Plot each equation: Identify the type of equation (e.g., linear, quadratic) and determine how it will appear on the graph. In our example, both equations are linear.
• Draw the lines accurately: Use the slope-intercept form or point plotting to sketch the lines. Consistent scales and a clean layout help in seeing intersections.
• Find the intersection: Look for where the graphs cross. The intersection point(s) provide the solution(s) for the system.

This method can be very visual and intuitive, particularly for systems with a single variable interaction or simple linear equations.

Parallel Lines

Parallel lines are lines in the same plane that never intersect. In the context of solving a system of linear equations, parallel lines indicate an inconsistent system with no solutions. Here's what students should know about them:

• Same slope, different intercepts: Parallel lines have equivalent slopes but different y-intercepts, making them always the same distance apart.
• No intersection point: Since these lines never meet, there is no corresponding x and y value that satisfy both equations simultaneously.
• Identifying parallel lines: Graphically, if lines run alongside each other without meeting, they are parallel. Algebraically, equal slopes suggest parallelism.

In our example, the two horizontal lines, \(y = 0\) and \(y = 4\), are parallel because they never touch, representing an inconsistent system.

Set Notation

Set notation is a mathematical language used to express groups of numbers or solutions. It's particularly useful for describing solution sets, especially when solving equations or systems. Here's a basic look at how it applies:

• Expressing solutions: When a system of equations has a solution, we list the solutions within curly braces. For instance, \( \{ (x, y) \text{ pairs} \} \).
• No solution: When no intersection exists between the graphs, as is evident with parallel lines, the solution set is empty, denoted by \(\emptyset\) or equivalently, \( \{ \} \).
• Infinite solutions: If solutions are infinite, which occurs when the same line is represented twice in different equations, we describe the set of all points on the line.

In our exercise, as the lines do not intersect, the solution set is the empty set, expressed in set notation as \( \emptyset \). This demonstrates the absence of any solutions.