Problem 56 - 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. $$\left\\{\begin{array}{l}-10 x+y=2 \\ -10 x+y=-3\end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations has no solution because the graphs of the equations are parallel lines with different y-intercepts.

Step 1: Write Equations in Slope-Intercept Form

We first need to rearrange each equation into the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. From the given system of equations, we have two equations: 1) -10x+y=2, which simplifies to y=10x+2,2) -10x+y=-3, which simplifies to y=10x - 3.

Step 2: Plot the Equations

Now we need to plot these two lines on the graph using a graphing utility. By comparing the y-intercepts and slopes, we'll be able to tell if the lines intersect, overlap or are parallel.

Step 3: Analyze the Graphs

After plotting the equations, we observe that the two lines are parallel and do not intersect or overlap. Two parallel lines have the same slope but different y-intercepts, hence they never intersect.

Key Concepts

Graphing Utilities

Graphing utilities are tools, often software or online applications, used for plotting mathematical equations on a graph. They are incredibly helpful in visualizing the relationships between equations and understanding systems of equations.

When working with graphing utilities, you input your equations and the software will automatically generate graphs. This is useful for complex calculations or when you need to quickly verify solutions. In this exercise, a graphing utility was used to determine the nature of the solution for the system of equations given.

To use graphing utilities effectively:

• Ensure your equations are properly formatted for input, often in slope-intercept form.
• Familiarize yourself with the graphing software's functions and settings.
• Use the zoom and pan features to closely analyze specific graph sections.

By using a graphing utility, you can easily identify if lines intersect, overlap, or are parallel which significantly aids in solving systems of equations.

Slope-Intercept Form

The slope-intercept form of a line is an equation expressed as \(y = mx + b\), where \((m)\) represents the slope of the line and \((b)\) represents the y-intercept. This form is particularly useful because it allows for quick graphing and understanding of the line's behavior.

To convert any linear equation to the slope-intercept form:

• Isolate the \(y\) variable on one side of the equation.
• Rearrange any other terms to clearly identify the slope \((m)\) and y-intercept \((b)\).

In the given system:

• The equation \(-10x + y = 2\) converts to \(y = 10x + 2\).
• The equation \(-10x + y = -3\) converts to \(y = 10x - 3\).

Utilizing the slope-intercept form helps reveal important characteristics of lines, such as their steepness and the point where they cross the y-axis.

Parallel Lines

Parallel lines are lines in the same plane that never intersect, no matter how far they are extended. They share the same slope but have different y-intercepts. This characteristic differentiates them from coincident lines, which also share the same slope but have the same y-intercept.

To determine if two lines are parallel, examine their equations written in slope-intercept form. If both equations have an identical slope \((m)\) but different y-intercepts \((b)\), they are confirmed to be parallel.

In this exercise, both lines had a slope of 10, but different y-intercepts, 2 and -3, respectively. This demonstrates they are parallel and consequently do not intersect, resulting in no solutions for the system of equations. Understanding these properties is key to analyzing and solving graphical problems involving systems of equations.