Problem 38 - Introductory Algebra for College Students

Question

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}x=3 \\ y=5\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the given system of equations is the set \(\{(3, 5)\}\).

Step 1: Graph the First Equation

Begin by drawing the graph of the first equation, \(x=3\). This is a vertical line that crosses the x-axis at the point (3, 0).

Step 2: Graph the Second Equation

Next, plot the graph of the second equation, \(y=5\). This is a horizontal line that crosses the y-axis at the point (0, 5).

Step 3: Identify the Intersection Point

The intersection point of the graphs is the solution of the system. In this case, the lines intersect at the point (3, 5). This is the only solution to this system of equations.

Step 4: Express the solution in set notation

The solution set can be represented in set notation as \(\{(3, 5)\}\). This indicates that the set contains one order pair, (3, 5), which is the solution of the system.

Key Concepts

Graphing Method

The graphing method is a visual technique to find the solutions of a system of equations. This approach involves plotting each equation on a graph to see where they intersect. To use this method effectively, it's important to understand:

• How to draw graphs accurately for each equation.
• The types of lines that represent different equations (e.g., linear equations will form straight lines).

For instance, in the given exercise, we have two equations: a vertical line where all points have the same x-coordinate but different y-values (\( x = 3 \)). You plot this by drawing a line that runs from top to bottom through the x-axis at 3.
Next, the horizontal line of the form \( y = 5 \), where every point on the line has the same y-value, is plotted by drawing a line through the y-axis at 5 that stretches left to right. These steps are crucial in creating a visual representation that leads you to find the intersection point.

Intersection Point

The intersection point is crucial in solving a system of equations using the graphing method. It's the point where two lines on a graph meet, representing the values of variables that satisfy both equations simultaneously. In the context of graphing equations:

• The intersection point provides the solution to the system of equations.
• It's determined by locating the exact point where the lines, representing each equation, cross each other.

In our exercise, the intersection point for the lines \( x = 3 \) and \( y = 5 \) is at the coordinates (3, 5). This location means that the values \( x = 3 \) and \( y = 5 \) satisfy both equations simultaneously. Finding this point confirms that there's a unique solution, ensuring that the system of equations is consistent and independent.

Set Notation

Set notation is a method of expressing mathematical solutions in a precise and standardized form. When dealing with solutions to systems of equations, set notation is used to describe all the solutions comprehensively. This format is particularly useful when:

• You want to specify that a particular ordered pair is a solution.
• The solution consists of all points that make both equations true.

In our example, once we identified that the point (3, 5) is the solution, we use set notation to express it. This is depicted as \( \{(3, 5)\} \), where the curly braces denote a set and the parentheses inside indicate an ordered pair. The set contains exactly one element, underscoring that there's a single solution to the system of equations. Utilizing set notation ensures clarity and academic precision in presenting mathematical results.