Problem 5 - Precalculus: Mathematical for Calculus

Question

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{aligned} -x+\frac{1}{2} y &=-5 \\ 2 x-y &=10 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has infinitely many solutions.

Step 1: Rewrite each equation in slope-intercept form

The first equation is \(-x + \frac{1}{2}y = -5\). To write this in slope-intercept form, solve for \(y\):\[\frac{1}{2}y = x - 5\]Multiply every term by 2 to clear the fraction:\[y = 2x - 10\]The second equation is \(2x - y = 10\). Solve for \(y\):\[-y = -2x + 10\]Multiply by -1 to get:\[y = 2x - 10\].Both lines have been rewritten in slope-intercept form: \(y = 2x - 10\).

Step 2: Graph the equations

Both equations are \(y = 2x - 10\), so graph this line on the coordinate plane. Start at the y-intercept (0, -10) and use the slope 2 (which means up 2, to the right 1) to plot a few more points in order to draw a straight line through these points on the graph.

Step 3: Analyze the graph for intersections

Notice that both equations are the same line, so they overlap completely on the graph. This means, for every point on line \(y = 2x - 10\), both equations hold true.

Step 4: Determine the number of solutions

Since the lines are identical and overlap entirely, the system has infinitely many solutions. This happens whenever both equations describe the same line in the coordinate plane.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves creating a visual representation of linear relationships on a coordinate plane. This process helps us understand how different lines relate to one another. In this exercise, we needed to graph the equations to identify the type of solution the system possesses. To do this, we follow a few steps:

• Rewrite the equations in slope-intercept form to make them easy to graph.
• Identify key elements such as slope and y-intercept.
• Use these elements to plot the equations on a coordinate plane.

By graphing, we can visually assess whether the lines intersect at a single point, are parallel (no intersection), or are exactly the same line, indicating infinitely many solutions.

Slope-Intercept Form

Understanding the slope-intercept form is crucial when dealing with linear equations. This form is expressed as \( y = mx + b \), where:

• \(m\) represents the slope of the line.
• \(b\) represents the y-intercept, which is where the line crosses the y-axis.

Rewriting equations into this form provides a straightforward method to graph them. It makes it easy to identify the starting point \(b\) on the y-axis and the direction \(m\) in which the line proceeds. In our exercise, both equations converted to \( y = 2x - 10 \), making it easy to see that they are exactly the same line.

Infinitely Many Solutions

A system of linear equations is said to have infinitely many solutions when the equations represent the same line. This means every point on the line is a solution to the system. In our example, both equations had the same slope and y-intercept after being rewritten to slope-intercept form. Thus, they represent the same line, confirming the presence of infinitely many solutions.When graphed, such equations will overlap completely, indicating that there are not just one, but an infinite number of solutions that satisfy both equations. Every coordinate \( (x, y) \) on that line fits either equation perfectly.

Coordinate Plane Analysis

Analyzing a coordinate plane allows us to deduce relationships between lines. This involves:

• Checking intersections: Are there points where the lines cross?
• Comparing slopes: Do the lines run parallel or intersect at some point?
• Observing parallels: Are the slopes the same, but the lines different?

For this exercise, upon plotting the linear equations, it was clear both lines were identical. Hence, the lines overlapped entirely, leading us to conclude the system has infinitely many solutions. Such analysis aids in understanding the principles that dictate linear systems and their solutions.