Problem 42 - 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. \(\left\\{\begin{array}{l}2 x+5 y=-4 \\ 3 x-y=11\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the given system of equations is \(x = 3\) and \(y = -2\). The system has exactly one solution.

Step 1: Solution Isolation

First, isolate one of the variables in one of the equations. Let's do it for the second equation, solving for \(y\): \(y = 3x - 11\).

Step 2: Substitution

Substitute this expression for \(y\) from equation 2 into equation 1: \(2x + 5(3x - 11) = -4\). This simplifies to: \(2x + 15x - 55 = -4\) or \(17x = 51\).

Step 3: Find \(x\)'s Value

Solving the equation \(17x = 51\) yields: \(x = 3\).

Step 4: Solve for \(y\)

Substitute \(x = 3\) into the solved equation from Step 1 to find the corresponding value of \(y\): \(y = 3(3) - 11 = 9 - 11 = -2\).

Key Concepts

Method of Substitution

When it comes to solving a system of linear equations, the method of substitution can be an intuitive and straightforward approach. The goal is to first isolate one of the variables in one of the given equations. Once that's been managed, the expressions outlining the isolated variable can be substituted into the other equation(s).

This process transforms the original system into a single equation with one variable, which can usually be solved with basic algebraic techniques. For example, if we have two equations, \(y = 3x - 11\) and \(2x + 5y = -4\), we substitute the expression for \(y\) from the first into the second equation. This yields one equation with one variable, \(2x + 15x - 55 = -4\), which simplifies to \(17x = 51\), and from there we can find the value of \(x\).

Set Notation

Set notation serves as a mathematical language to describe collections of elements, and it's particularly handy for representing the solution sets of equations or systems. In the context of linear systems, the solution can be expressed as an ordered pair, such as \( (3, -2) \) for the given system. But when we encounter a system with no solution or infinitely many solutions, set notation becomes invaluable.

For systems with no solution, we write \( \text{\emptyset} \) (the empty set), because there is no ordered pair that satisfies both equations. Conversely, in cases of infinite solutions, the set notation would involve a parameter to indicate a line or plane of solutions, such as \( \{ (x, 3x - 11) \mid x \in \mathbb{R} \} \) for a line of solutions where any real number \(x\) is possible.

No Solution and Infinite Solutions

When solving linear systems, we can encounter two special cases: systems with no solution and systems with infinitely many solutions. These situations arise based on the relationship between the lines represented by the equations.

No solution occurs when the lines are parallel and never intersect; they have the same slope but different y-intercepts. In this case, the system is considered inconsistent and the set of solutions is empty, denoted by \( \text{\emptyset} \).

Infinite solutions imply the lines are coincident, meaning they are on top of each other, essentially the same line. This happens when both equations represent the same linear relationship in different forms. Here, every point on the line is a solution, so we describe the solution set using a parameter that represents an infinite set.

Algebraic Solutions

Algebraic solutions are the results obtained by applying algebraic methods to solve equations. For linear systems, these solutions typically take the form of ordered pairs that satisfy all the given equations. They represent the intersection point of the lines or planes in a graph. The process includes techniques such as substitution, elimination, or graphing.

Continuing with our example, once the value of \(x\) is determined as \(x = 3\), we can substitute it back to find \(y\), leading to \(y = -2\). Thus, the algebraic solution for our system is the ordered pair \( (3, -2) \), which is the point of intersection of the two lines. This solution confirms that the system of equations is consistent and independent, having a unique solution.