Problem 18 - Algebra and Trigonometry

Question

\(15-20\) m 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. $$ \left\\{\begin{aligned} 2 x+6 y &=0 \\\\-3 x-9 y &=18 \end{aligned}\right. $$

Step-by-Step Solution

Answer

No solution; the lines are parallel.

Step 1: Rewrite the System in Slope-Intercept Form

First, we'll convert both equations to the slope-intercept form, which is \(y = mx + b\). For the first equation \(2x + 6y = 0\), solve for \(y\): \[6y = -2x\]\[y = -\frac{1}{3}x\]. For the second equation \(-3x - 9y = 18\), solve for \(y\): \[-9y = 3x + 18\]\[y = -\frac{1}{3}x - 2\].

Step 2: Graph the Equations

Use graph paper or graphing software to plot both lines. The first equation \(y = -\frac{1}{3}x\) is a line that passes through the origin with a slope of \(-\frac{1}{3}\). The second equation \(y = -\frac{1}{3}x - 2\) is parallel to the first line but shifts down 2 units on the y-axis.

Step 3: Analyze the Graph

Once both lines are graphed, observe that they are parallel and do not intersect. This is because they have the same slope \(-\frac{1}{3}\) but different y-intercepts (0 and -2 respectively). Parallel lines that never intersect indicate the system is inconsistent and has no solution.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to write the equation of a line. It is expressed as \( y = mx + b \), where:

• \( y \) is the dependent variable or output.
• \( m \) represents the slope of the line. It shows how steep the line is.
• \( x \) is the independent variable or input.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Breaking this down, the slope \( m \) can be understood as "rise over run," which indicates the change in \( y \) for a unit change in \( x \). This makes it easy to graph a line on a coordinate plane.
For instance, in the system given:

• The equation \( 2x + 6y = 0 \) becomes \( y = -\frac{1}{3}x \) once rewritten in slope-intercept form.
• The equation \(-3x - 9y = 18 \) transforms into \( y = -\frac{1}{3}x - 2 \).

Each equation has the form \( y = mx + b \), making it seamless to visualize and graph.

Parallel Lines

In geometry, parallel lines are lines in the same plane that never meet, no matter how far they are extended.
They have crucial characteristics:

• Parallel lines share the same slope. If two lines have the same slope, they are parallel.
• Even though they have the same slope, they have different y-intercepts. This means they are shifted vertically from each other.

In our exercise, both lines have a slope of \( -\frac{1}{3} \), making them parallel.
The line \( y = -\frac{1}{3}x \) intersects the y-axis at 0, while the line \( y = -\frac{1}{3}x - 2 \) intersects at -2. This shows they are shifted two units apart vertically, ensuring they never intersect. Understanding parallel lines is vital in identifying and analyzing the type of system a set of equations forms.

Inconsistent Systems

An inconsistent system is a system of equations with no solutions.
In terms of linear systems, this occurs when the lines are parallel and distinct, meaning they never intersect. Since solutions to systems often represent points of intersection, parallel lines indicate there is no shared point.

• For a system to be inconsistent, the equations must have the same slope (parallel lines) but different y-intercepts.
• Graphically, this is shown by lines that continue infinitely in both directions without meeting.

In our specific exercise, the system given is inconsistent since:

• Both lines have a slope of \( -\frac{1}{3} \).
• They have different y-intercepts: 0 for the first line and -2 for the second.

This mismatch in y-intercepts means the lines are distinct and do not share any points, clearly marking the system as inconsistent.