Problem 5 - Algebra and Trigonometry

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. $$\left\\{\begin{array}{l}{-x+\frac{1}{2} y=-5} \\ {2 x-y=10}\end{array}\right.$$

Step-by-Step Solution

Answer

The system has infinitely many solutions since the lines are identical.

Step 1: Identify the System Equations

The system of linear equations is given as: 1. \(-x + \frac{1}{2}y = -5\)2. \(2x - y = 10\)Our task is to graph these equations and find their intersection point, if it exists.

Step 2: Rearrange Equations to Slope-Intercept Form

Convert each equation to the slope-intercept form, which is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. **For Equation 1:**\(-x + \frac{1}{2}y = -5\)First, solve for \(y\):\[\frac{1}{2}y = x - 5\]\[y = 2x - 10\]**For Equation 2:**\(2x - y = 10\)Solve for \(y\):\[y = 2x - 10\]

Step 3: Graph the Equations

Both the equations result in \(y = 2x - 10\). Graph this line on the coordinate plane. As both lines are identical, they will overlap each other entirely.

Step 4: Analyze the Graph

Since both lines are identical, they cover one another completely. This means the lines have infinitely many points in common, which indicates that the system of equations has infinitely many solutions.

Key Concepts

Graphing Linear Systems

Understanding how to graph linear systems is fundamental in math. A linear system consists of two or more linear equations. In our case, we have two equations:

• \(-x + \frac{1}{2}y = -5\)
• \(2x - y = 10\)

To graph these equations, we convert them to a format that makes them easier to plot. Typically, we use the slope-intercept form for this purpose. Once in slope-intercept form, each equation gives a straight line when graphed on a coordinate plane.
Our solution lies at the point where these lines intersect. This intersection represents the values of \(x\) and \(y\) that satisfy both equations simultaneously. By checking the graph, we can tell if there is one solution, no solution, or infinitely many solutions.

Slope-Intercept Form

Converting equations to slope-intercept form \(y = mx + c\) is a critical step in graphing. The slope \(m\) provides information on how steep the line is, while the y-intercept \(c\) tells us where the line crosses the y-axis.
For our equations, we need to solve for \(y\) to rewrite them in this form:
**Equation 1:** \(-x + \frac{1}{2}y = -5\)\[\Rightarrow y = 2x - 10\]**Equation 2:** \(2x - y = 10\)\[\Rightarrow y = 2x - 10\]After converting the equations, we see they have the same slope and y-intercept, indicating they are the same line. This happens when the equations are dependent, meaning every point on the line is a solution.

Solution of Linear Systems

Solutions of linear systems tell us where two or more equations intersect.
Typically, a system can have:

• **One Solution**: The lines intersect at exactly one point.
• **No Solution**: The lines are parallel and never intersect.
• **Infinitely Many Solutions**: The equations represent the same line.

In our given system of equations, both equations eventually became \(y=2x-10\). This means they form a single line, or graphically overlap. Thus, every point on this line is a solution. Therefore, the system has infinitely many solutions, indicating that both equations are "infinite twins" in this scenario.