Problem 14 - Introductory Algebra for College Students

Question

Solve each system by the substitution method. If there is no solution or an infinite number of solutions, so state. Use set notation to express 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, stated in set notation as \(\emptyset\).

Step 1: Isolate one variable in one equation

From the second equation, \(\y=2-3x\), \(\y\) has been isolated.

Step 2: Substitute into other equation

Substitute \(y = 2-3x\) into the first equation, where it replaces y. Thus, the equation becomes: \(6x + 2(2-3x) = 7\).

Step 3: Solve for one variable

After simplifying the equation from Step 2, solve for \(x\): \n 6x + 4 - 6x = 7 \n 4=7 \n There is no solution as 4 does not equal to 7.

Step 4: Testing for Consistency

Since there is no solution for \(x\), hence the system is inconsistent and does not have any solution. So, there is no need to substitute the value of \(x\) back to the other equation to solve for \(y\).

Key Concepts

Understanding Systems of Equations

Systems of equations involve solving problems with two or more equations that are related because they share variables. In our example, the system is composed of two equations: \(6x + 2y = 7\) and \(y = 2 - 3x\). The aim is to find values for \(x\) and \(y\) that satisfy both equations simultaneously.

• The "substitution method" is a common approach for solving these systems. It involves isolating one variable in one of the equations, then substituting that expression into the other equation.
• This method can be very effective when one of the variables is easily isolated, as with the second equation \(y = 2 - 3x\) in our example.

When using substitution, the goal is to simplify the system to one equation with one variable. From there, you solve for that variable, and then back-substitute to find the other variable. Inconsistent systems might arise during this process, indicating no shared solution exists.

Recognizing Inconsistent Systems

An inconsistent system occurs when two or more equations within the system do not have any points in common. In other words, there is no set of values for the variables that satisfies all equations simultaneously. For instance, parallel lines represent such a system because they never meet.
In our problem, upon substituting \(y = 2 - 3x\) into \(6x + 2y = 7\), we get \(6x + 2(2 - 3x) = 7\). After simplifying, this equation reduces strangely to \(4 = 7\), which is clearly false. This means no values of \(x\) and \(y\) will ever satisfy both equations.

• Inconsistent systems often result in a contradiction (as we saw with \(4 = 7\)).
• This contradiction indicates that these lines are parallel and never intersect.
• Such a system has no solution and is classified as inconsistent.

Utilizing Set Notation for Solution Sets

Set notation is a useful mathematical language used to describe collections of solutions or answers. It is valuable because it provides a clear and concise way to specify solution sets, especially for complex equations.
When a system of equations is consistent, set notation might indicate the exact solution pairs \((x, y)\) like \( \{ (x, y) \} \). However, in the case of an inconsistent system, as we encountered here, the solution set is empty.

• The empty set is denoted by \(\varnothing\) or by \( \{ \} \), indicating that there are no solutions.
• This conveys that within the context of the given equations, no shared values for the variables exist.
• Using set notation helps to clearly communicate this lack of solutions in a universally understood format.

Overall, set notation serves as a precise method for expressing the outcomes of algebraic problems, either by detailing specific solutions or declaring none exist for inconsistent systems.