Problem 77 - Introductory Algebra for College Students

Question

Read Exercise \(72 .\) Then use a graphing utility to solve the systems. $$\left\\{\begin{array}{c}3 x-y=5 \\ -5 x+2 y=-10\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is the intersection point of the two lines. The graphing utility will find this intersection and provide the values for \(x\) and \(y\).

Step 1: Rewrite the Equations in Slope-Intercept Form

First, rewrite the equations into slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The system of equations becomes: \[y = 3x - 5\] and\[y = 2.5x + 5\]

Step 2: Graph the Equations

Then graph these two equations using the graphing utility. This will result in two straight lines.

Step 3: Find the Intersection Point

Find the intersection point of the two lines. This is the solution to the system of equations. Most graphing utilities offer a feature that allows you to find this point automatically.

Step 4: Interpret the Intersection Point

The intersection point represents the values of \(x\) and \(y\) that satisfy both equations at the same time. The x-coordinate of this point is the value of \(x\) and the y-coordinate is the value of \(y\) in the solution.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing linear equations. It is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, or where the line crosses the y-axis. This format is exceptionally useful because it allows us to easily identify and graph a line by understanding its slope and intercept.

• The slope, \( m \), describes how steep the line is. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
• The y-intercept, \( b \), tells us the point where the line will cross the y-axis.

In solving systems of equations, converting them into this form simplifies the process of graphing and finding the intersection point where the two equations meet.

Intersection Point

The intersection point of two lines in a system of equations is a crucial concept. It is the point where both lines meet on the graph, indicating the combination of \( x \) and \( y \) values that satisfy both equations simultaneously. When graphed, this point provides a visual representation of the solution to the system. Finding the intersection point:

• Observe where the two lines cross on the graph.
• The coordinates at the intersection give the values of \( x \) and \( y \).

In the given example, after plotting both equations from their slope-intercept forms, the intersection point indicates the solution to the original problem. This is often the primary goal when solving systems graphically.

Graphing Utility

A graphing utility can be a fantastic tool when working with systems of equations. It helps visualize the relationships between lines and makes it easy to determine the intersection point. Using a graphing utility, you can:

• Input equations in slope-intercept form.
• Automatically plot straight lines for each equation.
• Use built-in features to find the intersection point, making it quicker and more accurate.

These tools are invaluable in validating algebraic solutions and providing a concrete visual reference, especially useful in educational contexts for strengthening understanding.