Problem 38 - College Algebra

Question

In Exercises \(31-42,\) 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. \(4 x-2 y=2\) \(2 x-y=1\)

Step-by-Step Solution

Answer

The system of equations has infinitely many solutions, all pairs (x, y) that satisfy the equation \(4x - 2y = 2\)

Step 1: Rearrange the Equations

We have been given two equations. Let's rearrange them in the form \(ax + by = c\): \n Equation 1: \(4x - 2y = 2\) \n Equation 2: \(2x - y = 1\)

Step 2: Try Equating the Coefficients

We can multiply Equation 2 by 2 so that it matches the format of Equation 1: \n This gives us the new Equation 2: \(4x - 2y = 2\)

Step 3: Compare the Equations

Now, comparing both equations, we realize both equations are identical.

Step 4: Determine Type of Solution

Since both equations are identical, it means every solution to one of them is also a solution to the other, so there are infinitely many solutions. It means all pairs \((x, y)\) that satisfy the equation \(4x - 2y = 2\) are the solutions to this system.

Key Concepts

Infinite Solutions

When we talk about infinite solutions in the realm of a system of linear equations, we're referring to a scenario where there are endless pairs of values that satisfy all equations within the system simultaneously. This situation arises when the equations describe the same line or overlap each other perfectly in the case of two-dimensional geometry.

An important characteristic of such a system is that after simplification or manipulation of the equations, they appear identical. This is a clear indication that there is no single, unique solution but rather an infinity of solutions. To visualize it, you can imagine an infinite number of points lying on a line – each point represents a valid solution pair for the variables involved. This concept is crucial in understanding that not all systems will have a single solution or no solution at all; some have infinitely many, which we see exemplified in our textbook exercise.

System of Linear Equations

A system of linear equations consists of two or more linear equations that we aim to solve simultaneously – that is, to find the values for the variables that make all equations true at the same time. Typically, these systems are comprised of equations in the form of \(ax + by = c\), where 'a', 'b', and 'c' are constants and 'x' and 'y' are the variables.

Systems of linear equations can have a single solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (the same, overlapping line). The goal is to determine which of these scenarios is true for the equations presented. In our case, we have identified that the system has infinitely many solutions, meaning the two lines represented by the equations are coincident.

Set Notation

Set notation is a method of describing collections of elements or numbers, and it's particularly useful in conveying the solutions to systems of equations in a concise and standardized form. In the context of infinite solutions, set notation can express complex ideas like all the points that lie on a certain line. Here's how we might express the infinite solutions to our textbook problem: \[ \left\{ (x, y) | 4x - 2y = 2 \right\} \].

This states that the set includes all pairs of \((x, y)\) that satisfy the given linear equation. Set notation's power lies in its ability to encapsulate an endless list of solutions in a precise and mathematical way, ensuring clarity and consistency in communication.

Method of Substitution

The method of substitution is one of the strategies to solve systems of linear equations, which involves isolating one variable in one equation and substituting it into the other. This process can simplify the system to a single equation with one variable, which can then be solved.

However, in the exercise provided, we observed that after manipulating the equations, they became identical, indicating a scenario of infinite solutions and rendering substitution unnecessary. If they weren't identical, we'd isolate 'y' in the equation \(2x - y = 1\), for instance, to get \(y = 2x - 1\) and substitute this expression for 'y' in the other equation, proceeding with finding 'x', and thereafter 'y'. This method provides a systematic approach to resolving the values of variables involved in the system.