Problem 43 - 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. \(\left\\{\begin{array}{r}x+3 y=2 \\ 3 x+9 y=6\end{array}\right.\)

Step-by-Step Solution

Answer

The system of equations has infinitely many solutions, which are all the points on the identical line represented by the equations. These can be represented as \{(x,y) | x + 3y = 2\}.

Step 1: Recognize the Form of the System

The two equations are: \(x+3y=2\) and \(3x+9y=6\). These are of the form \(Ax+By=C\). Notice that the second equation can be derived by multiplying the first equation by 3.

Step 2: Divide The Second Equation

To make the relationship clear, divide every term in the second equation by 3 to get \(x+3y=2\). Now it is clear the two equations are identical.

Step 3: Identify the Solution

Since the two equations are identical, they represent the same line. This implies that the system of equations has infinitely many solutions, which are all the points on the line itself. This can be represented in set notation as \{(x,y) | x + 3y = 2\}

Key Concepts

Method of Solving Systems

When faced with a system of linear equations, such as the one provided in the exercise, there are various methods one could choose to find the solutions. Among these are the graphing method, substitution method, elimination method, and matrix approaches like Gauss-Jordan elimination and using row-reduction.

However, before choosing a method, it is critical to assess the structure of the given equations. In the exercise, the equations seemed to be multiples of each other. When you encounter a system where one equation is a multiple of another, you can anticipate specific outcomes regarding the solutions to the system. If after simplification or manipulation both equations become identical, as in our exercise, it indicates that there is not just a single solution but rather infinitely many solutions, as both equations represent the same line.

No Solution Identification

Identifying systems with no solution is an essential skill when solving systems of equations. A system has no solution when the lines represented by the equations are parallel and do not intersect at any point. This scenario is recognizable algebraically when the two equations, after being simplified, result in a contradiction, such as a statement that equates two non-equal constants (e.g., '5 = 3'). In such cases, the system is inconsistent and it can be stated that there are no points (x, y) that satisfy both equations simultaneously. Recognizing patterns, like parallel lines, which lead to no solutions helps to avoid unnecessary computation and enables a more strategic approach to problem solving.

Infinitely Many Solutions

A system of equations has infinitely many solutions when the two equations represent the same line. This means every point on the line is a solution to the system. In terms of algebra, this occurs when one equation can be transformed into the other by multiplying or dividing by a non-zero constant, or through other algebraic manipulations.

In the given exercise, dividing the second equation by 3 transformed it into the first equation, revealing that they are, in fact, the same line, hence the infinite number of solutions. For students understanding this concept, it's akin to stacking two identical objects; they align perfectly at every point, which symbolizes our infinite solutions along the line of the equation.

Set Notation

Understanding set notation is vital when dealing with systems of equations, especially when expressing the solution of a system. In set notation, the solutions of a system are described as a set of ordered pairs that satisfy all equations within the system.

The set notation for the solution of a system that has infinitely many solutions, like the one in our exercise, includes the use of the vertical bar '|' to denote 'such that'. The set looks like this: \( \{(x,y) | x + 3y = 2\} \), illustrating all the points (x, y) that satisfy the equation. Embracing set notation helps in clearly communicating the complete set of solutions to a system and is a fundamental tool in higher mathematics.