Problem 19 - Precalculus Mathematics for Calculus

Question

Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether 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

Infinitely many solutions, as the lines are identical.

Step 1: Convert Equations to Slope-Intercept Form

The goal is to write each equation in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.For the first equation \(-x + \frac{1}{2}y = -5\), solve for \( y \):\[ \frac{1}{2}y = x - 5 \]\[ y = 2x - 10 \]For the second equation \(2x - y = 10\), solve for \( y \):\[ -y = -2x + 10 \]\[ y = 2x - 10 \]

Step 2: Graph the Equations

Plot each equation on a coordinate plane using their slope-intercept forms:Both equations are \( y = 2x - 10 \). This means they have the same slope and the same y-intercept. As a result, each line will overlap completely when graphed.

Step 3: Analyze the Graph

Since the two lines are identical when plotted, this implies they overlap entirely, indicating they are the same line.

Step 4: Determine the Solution Type

Because the two equations graph as the same line, the system does not have a single unique solution, nor does it have no solution. Instead, it has infinitely many solutions because every point on the line is a solution to the system.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in understanding linear systems. A linear equation is an equation that forms a straight line when plotted on a graph. In order to graph a linear equation, it is often helpful to rewrite the equation in slope-intercept form, which is easier to interpret and plot.
When visualizing a linear equation:

• Identify two essential components: the slope and the y-intercept.
• Start plotting the graph by marking the y-intercept on the y-axis.
• Use the slope, which indicates the rise over the run, to determine the direction and steepness of the line.
• Draw a straight line through these points to extend across the grid.

Graphing helps to visually determine the relationship between sets of lines, and ultimately find solutions for linear systems.

Slope-Intercept Form

The slope-intercept form of a line is represented as \( y = mx + b \), where:

• \( m \) is the slope of the line, showing how steep the line is and in which direction (upwards if positive, downwards if negative).
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

This form makes it straightforward to graph a line because:

• It provides a clear starting point on the graph at \( b \).
• It describes how to proceed from the intercept using the slope \( m \), calculated as \( \frac{\text{rise}}{\text{run}} \).

To convert an equation into slope-intercept form, solve the equation for \( y \) so that it is explicitly isolated on one side of the equation.
In the original exercise, transforming the equations into this form revealed that both lines were identical, leading towards understanding the solution of the system.

Solutions of Linear Systems

A linear system is a collection of one or more linear equations involving the same set of variables. Analyzing their graphs helps determine the type of solutions they have: one solution, no solution, or infinitely many solutions.
- **One Solution**: Occurs when two lines that graph as intersecting form a unique point; this point is the solution to the system. - **No Solution**: Happens when the lines are parallel and never intersect, implying they have the same slope but different y-intercepts. - **Infinitely Many Solutions**: Occurs when the lines are identical, meaning they overlap completely as seen in the original exercise. This indicates every point on the line satisfies both linear equations.
Determining the solution type starts by graphing each equation and observing their interaction. For the given problem, the overlapping graphs confirmed that every point on the line represents a solution, hence showing infinite solutions.