Problem 49 - Intermediate Algebra

Question

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) $$ \left\\{\begin{array}{l} y=3 \\ x=2 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has a unique solution at the intersection point (2, 3).

Step 1: Identify the Graph Type of Each Equation

The first equation is \( y = 3 \), which represents a horizontal line that crosses the y-axis at 3. The second equation is \( x = 2 \), which is a vertical line crossing the x-axis at 2.

Step 2: Plot the Lines on a Graph

On a coordinate plane, draw the horizontal line \( y = 3 \) by going from left to right at the level of \( y = 3 \). Then, draw the vertical line \( x = 2 \) by going from top to bottom at the line where \( x = 2 \).

Step 3: Find the Intersection Point

The intersection of the two lines occurs where both conditions are satisfied: \( y = 3 \) and \( x = 2 \). This point is the ordered pair (2, 3).

Step 4: Determine if the System is Consistent or Inconsistent

Since the lines intersect at the point (2, 3), the system is consistent, meaning there's a unique solution.

Key Concepts

Graphing Method

The graphing method is a visual way to solve a system of equations. By drawing each equation as a line on a graph, you can find their intersection. This intersection represents the solution to the system, proving where both equations are true at the same time.

To use the graphing method, follow these steps:

• Rewrite each equation in a form that easily shows how to plot them on a graph (either in slope-intercept form or as specific forms of horizontal and vertical lines).
• Plot each equation on a coordinate grid.
• Observe where the lines intersect. This point—if it exists—is the solution.

The graphing method provides a visual context, making it easier to understand the relationships between equations. It shows whether the system has a unique solution, no solution, or infinitely many solutions from the way the lines interact.

Consistent System

A consistent system of equations is one that has at least one solution. It occurs when the lines representing the equations intersect at a single point or overlap entirely—indicating infinite solutions.

Let's break down two types of consistent systems:

• **Independent Systems**: These have exactly one solution. In this case, the lines intersect at one point. The intersecting point provides the unique solution.
• **Dependent Systems**: These result when the lines are completely on top of each other, meaning they have infinite solutions. Here, every point on the line is a solution.

In our current exercise, the system is consistent and independent because the lines intersect exactly at one point, \((2, 3)\). This tells us that there's a unique solution where both equations hold true simultaneously.

Horizontal and Vertical Lines

Horizontal and vertical lines are special cases in graphing that can make finding solutions simpler. Understanding them is essential for solving systems graphically.

For horizontal lines:

• These are represented by equations of the form \( y = c \), where \( c \) is a constant. Every point on this line has the same y-value; hence, it runs left and right across the graph.
• In our exercise, \( y = 3 \) is a horizontal line passing through all points where the y-coordinate is 3.

For vertical lines:

• These are represented by equations of the form \( x = c \), where \( c \) is a constant. Every point on the line shares the same x-coordinate, so it goes up and down the graph.
• In our exercise, \( x = 2 \) is a vertical line passing through all points where the x-coordinate is 2.

The intersection of horizontal and vertical lines forms a clear-cut solution to a system of equations, which is simple to locate by eye.