Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to determine whether the system of equations has one solution, two solutions, or no solution. $$\left\\{\begin{aligned}-\frac{1}{2} x+y &=-1 \\ 7 x+y &=2 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The number of solutions will be determined by the interaction of the two plotted lines on the graph. It could either be one solution (if the lines intersect), two solutions (if the lines overlap), or no solution (if the lines are parallel). The exact answer will depend on the graph produced by the specific graphing utility used.

Step 1: Formulate the Equations for Graphing

Rewrite the equations in the form \(y = mx + c\), where \(m\) represents the gradient, \(c\) the y-intercept, \(x\) the x-coordinate, and \(y\) the y-coordinate. The first equation becomes \(y = 0.5x + 1\). The second equation is \(y = -7x + 2\).

Step 2: Plot the Equations

Using a graphing utility (like Desmos, GeoGebra, or even a graphing calculator), plot both equations on the same graph. It's standard practice to use different colors for each equation to avoid confusion.

Step 3: Analyze the Graph

Check how the graphs of the two equations interact. There are three scenarios to consider: the lines intersecting at a single point (one solution), the lines overlapping each other (infinite number of solutions or the system being dependent), or the lines are parallel having no intersections (no solution).

Key Concepts

Understanding Graphing Utilities

Graphing utilities are tools that help you visualize mathematical equations and functions. They are especially useful for solving systems of equations by graphing. When you graph equations, these utilities allow you to see how different lines interact. This is particularly helpful in determining the number of solutions a system of linear equations has. Some popular graphing utilities include:

• Desmos: An interactive online graphing calculator that's user-friendly and great for beginners.
• GeoGebra: A more advanced graphing tool that provides additional features and is useful for more complex equations.
• Graphing calculators: Devices that can plot graphs, solve equations, and perform various mathematical operations.

When using these tools, it's a smart approach to plot each equation in a different color. This makes it easier to track and analyze how the graphs overlap or intersect.

Exploring Linear Equations

Linear equations are equations of the first degree, which means they graph as straight lines. They have the general form of \( y = mx + c \), where:

• \( m \) is the slope or gradient of the line, indicating its steepness and direction.
• \( c \) is the y-intercept, which is the point where the line crosses the y-axis.

In our exercise, the linear equations involved are \( y = 0.5x + 1 \) and \( y = -7x + 2 \). The slope \( m \) tells us:

• If \( m > 0 \), the line is rising.
• If \( m < 0 \), the line is falling.
• If \( m = 0 \), the line is horizontal.

Understanding these components helps you quickly determine the shape and tilt of each line when graphing.

Solution Analysis for Systems of Equations

Analyzing the graph of a system of equations allows us to understand how many solutions exist. Depending on the interaction of the graphed lines, there are typically three possible scenarios:

• **One Solution**: The lines intersect at exactly one point. This point is the solution to the system of equations, where both equations are satisfied simultaneously.
• **No Solution**: The lines are parallel and do not intersect. This means there is no set of \( x \) and \( y \) values that satisfy both equations, and the system is inconsistent.
• **Infinite Solutions**: The lines overlap completely, meaning they are the same line. Every point on the line is a solution, and the system is dependent.

In our exercise, using a graphing utility revealed that the lines intersect at a single point, indicating that the system has one unique solution. This visual confirmation is both effective and straightforward, making graphing a practical method for solving systems of equations.