Problem 38 - 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. \(\left\\{\begin{array}{l}6 x+2 y=7 \\ y=2-3 x\end{array}\right.\)

Step-by-Step Solution

Answer

The system of equations has no solution, so its solution set is \(\emptyset\).

Step 1: Analyze the system

First, look at the system of equations. It can be seen that the second equation \(y=2-3x\) is already solved for \(y\). Now this equation can be substituted into the first one.

Step 2: Solve the first equation

Substitute \(y\) from the second equation \(y=2-3x\) into the first equation \(6x+2y=7\). You will get a new equation \(6x+2(2-3x)=7\). Now simplify and solve for \(x\): \[6x+4-6x=7\] \[-6x+6x=7-4\] \[0x=3\] Now, it is clear that the equation is false, so the system has no solution.

Step 3: Determine the solution set

The solution set for this equation is the empty set because there are no values of \(x\) and \(y\) that would satisfy both equations at the same time. Therefore, we can express the solution set in set notation as \(\emptyset\).

Key Concepts

Method of Substitution

One of the fundamental techniques in solving systems of equations is the method of substitution. This involves replacing one variable with an expression containing the other variable. It's an effective strategy when one equation in the system is already solved for a particular variable, making it easy to substitute into the other equation(s).

For instance, let's examine a system of two equations, where one is already solved for the variable, say, \(y = f(x)\). To apply the method of substitution, we take the expression for \(y\) and substitute it into the other equation where \(y\) appears. Once substituted, we are left with an equation in one variable that can then be simplified and solved. After finding the value of that variable, back substitution is used to find the value of the original variable.

No Solution in Systems of Equations

In algebra, not all systems of equations have a solution. A system that has no solution is said to be inconsistent. This usually occurs when the lines represented by the equations are parallel, hence, they never intersect. In our set example, after substituting and simplifying, we found the equation \(0x = 3\), which is a contradiction as it implies that \(0 = 3\). This contradiction means that there's no possible value of \(x\) that can satisfy the equation. Therefore, no solution exists for the system.

Identifying a system with no solution is crucial because it saves time; understanding that further calculations would be futile allows us to conclude and move on to other problems.

Set Notation

Set notation is a systemic way of defining collections of objects, and in algebra, sets are used to denote collections of solutions. For equations and inequalities, solutions can be described in interval or set-builder notation. When a system of equations has no solution, we use the empty set symbol, \(\emptyset\), to represent the solution set. It signifies that there is no element (or no point in the coordinate plane) that satisfies all equations in the system simultaneously.

The use of set notation in expressing solutions is concise and universally understood in the mathematical community, providing a clear and effective means of communication.

Algebraic Problem-Solving

Algebraic problem-solving involves finding the unknowns in equations and inequalities by using a systematic approach. It includes understanding the problem, devising a plan, and then carrying out that plan through manipulations like combining like terms, factoring, and applying arithmetic operations. In the context of a system of equations, the goal is to find a set of values that satisfies all equations at once. This often involves strategic steps like the application of substitution or elimination, graphing methods, or even more advanced techniques such as matrix operations in linear algebra.

Successful algebraic problem-solving hinges on the ability to recognize patterns, choose appropriate strategies, and logically infer the implications of each move within a set structure. In this way, algebra not only equips learners with a toolset for tackling equations, but also develops critical thinking and analytical skills.