Problem 32 - Algebra and Trigonometry

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 given system of equations has no solution.

Step 1: Substitute the value for \(y\) in the first equation

Substitute the expression \(2-3x\) for \(y\) in the first equation. This will yield a new equation: \(6x + 2(2 - 3x) = 7\)

Step 2: Simplify and solve the equation

Simplify the equation by expanding and then moving the terms involving \(x\) to one side and the constants to the other side: \(6x + 4 - 6x = 7\), this simplifies to \(4=7\)

Step 3: Check the result

Since \(4=7\) is a false statement, this means that the system of equations has no solution.

Key Concepts

No Solution System

In mathematics, a system of equations may have one solution, infinitely many solutions, or no solution at all. A no solution system, also known as an inconsistent system, is one where the equations represent parallel lines that never intersect. Consequently, since there are no points of intersection, there are no solutions that satisfy both equations simultaneously.

Let's analyze the problem at hand. We have two linear equations: the first is 6x + 2y = 7 and the second is y = 2 - 3x. Using substitution, we arrived at an equation 4 = 7 after simplification. This statement is a contradiction because it's not possible for the number 4 to equal 7; thus, we can state confidently that our original system of equations has no solution. When writing in set notation, we denote this as an empty set, symbolized by \(\emptyset\) or { }, signifying that there are no values of x and y that can satisfy both equations.

Substitution Method

The substitution method is a powerful technique utilized to find the solution to a system of equations. The core idea involves expressing one variable in terms of another using one equation, and then substituting this expression into the other equation.

In our exercise, we started by isolating y in the second equation, yielding y = 2 - 3x. Then, we substituted this expression in place of y in the first equation, resulting in a single equation with one variable: 6x + 2(2 - 3x) = 7. We followed algebraic steps to simplify this equation, searching for a point of intersection. While this method is straightforward when there is a solution, if we reach a contradiction, as in this case with 4=7, it indicates a no solution system. Substitution is especially effective for linear equations and can also be adapted for use with non-linear systems, though the latter might require additional techniques or refined algebraic manipulation.

Set Notation

To express solutions to systems of equations, set notation is employed as a standardized mathematical language. This notation is used to describe the collection of elements that satisfy the conditions of the problem. For instance, in a system with a unique solution, you would enclose that solution in curly braces, like {(x, y)}, with x and y being the values that solve both equations.

However, when dealing with a no solution system, set notation becomes quite simple: the solution is represented as an empty set, written as \(\emptyset\) or { }. This denotes that there are no elements—no pairs of x and y values—that meet the criteria defined by the system. In contrast, for systems with infinitely many solutions, you would express the solution set using a description that captures all the solution pairs, which often involves a parameter, like {(x, 2 - 3x) | x is a real number}. Set notation allows for a precise and concise way to depict the range of possibilities for the solutions to a given system.