Problem 39 - Thinking Mathematically

Question

Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(\left\\{\begin{array}{l}y=3 x-5 \\ 21 x-35=7 y\end{array}\right.\)

Step-by-Step Solution

Answer

The given system of equations represents coincidental lines and therefore has infinitely many solutions. The solution set in set notation is: S = { (x, y) | y = 3x - 5 for all real numbers x }.

Step 1: Analyze the System

First, examine the system of linear equations to check if there are equivalent or proportional. This will show if the lines represented by the equations are the same (coincident), parallel (no intersection), or intersect at a single point (one solution). In our problem, the two equations are: \[y=3x-5\] \[21x-35=7y\] Rewriting the second equation gives us \[y=3x-5\] which is identical to the first equation.

Step 2: Identifying the Relationship

Since the two equations are the identical, then they represent the same line. This means that these lines coincide with each other. A system of equations with coincidental lines always has infinitely many solutions, because any point on the line is a solution to both equations.

Step 3: Set Notation For the Solution

Every pair \((x, y)\) that satisfies the equation \(y = 3x-5\) will be a solution. Therefore, the solution can be represented in set notation as: S = \{ (x, y) | y = 3x - 5 for all real numbers x }\}

Key Concepts

Understanding Linear Equations

Linear equations are equations that make straight lines when graphed. They follow the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. These equations represent relationships where one variable changes at a constant rate with another.

In the original exercise, the equation \(y = 3x - 5\) is linear. Here, \(3\) is the slope, indicating the line rises 3 units for every 1 unit it moves to the right. The number \(-5\) is the y-intercept, showing the line crosses the y-axis at \(-5\).

Linear equations are foundational in algebra, helping us understand how variables relate to each other in a predictable way.

Exploring Infinitely Many Solutions

When a system of linear equations results in the same line, it has infinitely many solutions. This happens when both equations in the system describe the same line.

In such cases, every point on the line satisfies both equations. For example, the given system:

• \(y = 3x - 5\)
• \(21x - 35 = 7y\)

Becomes identical once simplified. This tells us that the two lines coincide. Since they are the same, any point that lies on this line is a solution to the system.

This shows how two seemingly different equations can end up describing the very same geometric line.

Using Set Notation for Solutions

Set notation is a mathematical way to describe a group of numbers or objects. For systems of equations with infinitely many solutions, set notation neatly expresses all possible solutions.

In our example, the solution is written in set notation:

\[ S = \{ (x, y) \mid y = 3x - 5 \text{ for all real numbers } x \} \]

This means any pair \((x, y)\) that satisfies \(y = 3x - 5\) forms part of the solution set \(S\). Set notation provides a clear and concise way to express that there are numerous solutions.

Coincident Lines Simplified

Coincident lines mean they lie on top of each other, representing the same geometric entity. This occurs when two lines in a system share the exact equation, even if they initially appear different.

By transforming \(21x - 35 = 7y\) into \(y = 3x - 5\), we see they describe the same line. Coincident lines always share every point, leading to infinitely many solutions.

Recognizing coincident lines helps simplify problem-solving. Instead of finding one intersection point, it's straightforward to understand that all points on the line solve both equations. Knowing this saves time and clarifies the nature of the solution.