Problem 49 - Introductory Algebra for College Students

Question

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{aligned} 3 x-2 y &=6 \\ y &=3 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x=4\) and \(y=3\).

Step 1: Identify the Known Variable

The system of equations given is: \[\begin{aligned} 3 x-2 y &=6, \ y &=3 \end{aligned}\] From the second equation, it can be seen that \(y=3\).

Step 2: Substitute the Value of the Variable

Put \(y=3\) into the first equation: \[3x -2(3) = 6\]. This simplifies to: \[3x -6 = 6\].

Step 3: Solving the Simplified Equation

Now solve \[3x -6 = 6\] for \(x\). This results in: \[3x = 12\] and then \[x = 4\].

Step 4: Verify the Solution

Substitute \(x=4\) and \(y=3\) into both original equations to ensure that both are satisfied. Substituting these values into the first equation results in \(3(4)-2(3)=6\), which holds true. Similarly, the second equation \(y=3\) also holds for \(y=3\). Therefore, the solution is verified.

Key Concepts

Substitution Method

The substitution method is a popular technique for solving systems of equations. It involves solving one of the equations for one variable and then substituting this expression into the other equation. This method is particularly useful when one of the equations is easily solved for one of the variables.
In the given exercise, the system of equations is:

• \(3x - 2y = 6\)
• \(y = 3\)

Here, the substitution method becomes straightforward because one of the equations already explicitly gives the value of \(y\) as 3. By substituting \(y = 3\) into the first equation, you convert a system of two equations with two variables into a single equation with one variable. This simplification makes it easier to find the value of \(x\).

Once you substitute \(y = 3\) into \(3x - 2y = 6\), you get \(3x - 6 = 6\). Solving this equation yields \(x = 4\). After solving for \(x\), substituting both \(x\) and \(y\) into either of the original equations helps verify the solution is correct. This simplicity is why the substitution method was an ideal choice for this exercise.

Linear Equations

Linear equations are equations where the highest power of the variable is one, making them distinctively straightforward to work with. The primary characteristic of linear equations is that they graph as straight lines.
In the given problem, both of the equations \(3x - 2y = 6\) and \(y = 3\) are linear. This means:

• Each equation describes a line on a two-dimensional plane.
• The solution to the system represents the intersection point of these two lines.

Understanding linear equations helps in predicting the behavior of solutions:- If two lines intersect at one point, there is exactly one solution.- If they are parallel (never meet), there is no solution.- If they are identical (coincident), there are infinitely many solutions.

In this exercise, substituting \(y = 3\) into the linear equation \(3x - 2y = 6\) simplifies the problem to a single linear equation with one unknown, illustrating the predictability and ease of handling linear systems.

Set Notation

Set notation is a mathematical tool used to precisely express the solutions of equations or systems of equations. It helps clearly communicate solution sets, which can be especially useful in systems of equations.
In set notation, the solution to the given system, once verified by substitution, is expressed as \((x, y) = (4, 3)\). This notation signifies a set containing a single ordered pair, which is the solution to the system. Here, it is represented as:

• \(\{(4, 3)\}\)

Set notation is beneficial because:- It provides clarity and precision in representing solutions.- It is versatile, capable of detailing no solutions, one solution, or infinitely many solutions.
For instance, if a system had no solutions, it might be written as \(\emptyset\) (the empty set). For an infinite number of solutions, a more complex expression using set or interval notation would describe the solution set.

In conclusion, set notation is an elegant way to convey the unique solution found by solving the system through substitution, highlighting the clear intersection point at \((4, 3)\) in the coordinate plane.