Problem 38 - College Algebra

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. \(\left\\{\begin{array}{l}4 x-2 y-2 \\ 2 x-y-1\end{array}\right.\)

Step-by-Step Solution

Answer

The system of equations has infinitely many solutions, represented in set notation as \[ \{ (x, y) \, | \, y = 2x-1, x \in R \}\]

Step 1: Identify the given equations

The given system of linear equations is \[\left\{\begin{array}{l}4 x-2 y-2=0 \\ 2 x-y-1=0\end{array}\right.\] It can be noticed that equation (2) is exactly half of equation (1). So, these are not independent equations, but rather the same equation written in different forms.

Step 2: Identify the type of solution

Since these are identical equations, they'll intersect at infinitely many points, i.e., they represent the same line. Therefore, the system of equations has infinitely many solutions.

Step 3: Express the solution in set notation

The solution can be presented in set notation by converting one equation to y = f(x) form, i.e., \[y = 2x - 1\] So the solution set is \[ \{ (x, y) \, | \, y = 2x-1, x \in R \}\] where R denotes the set of all real numbers.

Key Concepts

Method of Solution for Linear Systems

When dealing with systems of linear equations, there are several methods to find the solution, i.e., the set of values that satisfy all equations simultaneously. The most common methods include:

• Graphing: Plotting each equation on a graph to find the point(s) of intersection.
• Substitution: Solving one of the equations for one variable and then substituting the result into the other equation.
• Elimination (or Addition): Adding or subtracting the equations to eliminate one variable, making it easier to solve for the remaining one.
• Matrix method (Row Reduction or Gaussian Elimination): Transforming the system into a matrix and performing operations to solve for the variables.

Each method has its own pros and cons, and the choice of method can depend on the specific system at hand. For instance, graphing is a good visual representation but might not be precise for complex equations. Substitution and elimination are algebraically straightforward but can become cumbersome with larger systems. The matrix method is powerful for handling multiple variables and equations but requires understanding matrix operations.

The exercise provided is an example where the elimination method might be the most straightforward since the second equation is a multiple of the first. This indicates that we are dealing with dependent equations, which leads to special cases in the solution set.

No Solution and Infinitely Many Solutions

In algebra, systems of linear equations can have one of three possible types of solutions: a single solution, no solution, or infinitely many solutions. How do we distinguish between them?

Unique Solution

If the graphs of the equations intersect at exactly one point, the system has a unique solution, representing that specific point of intersection.

No Solution

If the graphs are parallel and never intersect, there will be no set of values that satisfy both equations simultaneously, indicating no solution.

Infinitely Many Solutions

The case of infinitely many solutions occurs when the equations represent the same line. Any point on this line is a solution to the system, hence there are infinitely many solutions.

In the given exercise, the conclusion that the system has infinitely many solutions comes from recognizing that both equations are proportional (the second is half of the first), indicating they are the same line. When asked for this type of solution in set notation, we must first rearrange the equation into a function of one variable and then describe all the points on the line. This involves indicating that one variable is free to take any value within a certain set, usually the set of all real numbers.

Set Notation in Algebra

Set notation is a standardized mathematical language used to describe collections of elements. In the context of algebra, we use set notation to specify the solution sets of equations or systems of equations.

When we express a solution set in algebra, we typically use braces \( \{ \} \) to collect all the solution elements which satisfy the given conditions. For example, if a solution to a single variable equation is any real number that's greater than 2, we would write this as \( \{ x \in \mathbb{R} \, | \, x > 2 \} \), where \( \in \) denotes 'element of', \( \mathbb{R} \) represents the set of real numbers, and the vertical bar \( | \) means 'such that'.

In the step-by-step solution provided, the set notation \( \{ (x, y) \, | \, y = 2x-1, x \in \mathbb{R} \} \) reads as 'the set of all pairs of real numbers \( (x, y) \) such that \( y \) equals \( 2x-1 \)’. This perfectly expresses the infinitely many solutions condition as it includes all xy-pairs lying on the line defined by the equation y = 2x - 1.