Problem 4 - 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. $$\left\\{\begin{aligned} 2 x+6 y &=0 \\\\-3 x-9 y &=18 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has no solution, as the lines are parallel with different y-intercepts.

Step 1: Rewrite Equations in Slope-Intercept Form

The given system of equations is: \(2x + 6y = 0\) and \(-3x - 9y = 18\). First, solve each equation for \(y\) to put them into the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.\1. Solving the first equation: \(2x + 6y = 0\) gives \(6y = -2x\). Then, \(y = -\frac{1}{3}x\).\2. Solving the second equation: \(-3x - 9y = 18\) leads to \(-9y = 3x + 18\). Thus, \(y = -\frac{1}{3}x - 2\).

Step 2: Graphing Each Line

Now that both equations are rewritten, graph them to find their intersection.\- The first line \(y = -\frac{1}{3}x\) passes through the origin with a slope of \(-\frac{1}{3}\). One point is (0,0).\- The second line \(y = -\frac{1}{3}x - 2\) has the same slope \(-\frac{1}{3}\) and crosses the y-axis at -2. One point is (0,-2) and another might be found by moving with the slope (3 units right and 1 unit down, to (3,-3)).

Step 3: Analyze the Graph

Upon graphing both lines, note that they have the same slope but different y-intercepts. This means the lines are parallel and will never intersect. Because the lines do not intersect, the system of equations has no solution.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental technique for visually representing solutions of linear systems. This involves plotting points on a coordinate grid. When graphing, each equation in a linear system is represented by a line. The points on the line are solutions that satisfy the equation. To begin this process, you'll need to transform each equation into a format that's easy to graph, typically slope-intercept form (more on this later). Once in this form, you utilize each line's slope and y-intercept to draw its respective line.

• The slope dictates how steep the line is.
• The y-intercept indicates where the line crosses the y-axis.

By carefully plotting these lines, you can examine where (or if) they intersect, which determines the solution to the linear system.

Slope-Intercept Form

The slope-intercept form of a linear equation is useful for easily graphing lines. It is represented as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope \( m \) tells you the direction and steepness of the line. If \( m \) is positive, the line rises; if negative, it falls. The y-intercept \( b \) is the point where the line crosses the y-axis, providing a starting point for drawing the line.
The first step in working with a linear system usually involves transforming each equation into this form. For instance, when given the equation \( 2x + 6y = 0 \), solving for \( y \) yields \( y = -\frac{1}{3}x \). This shows us:

• The slope is \(-\frac{1}{3}\)
• The line crosses the y-axis at the origin (0,0)

Parallel Lines

Parallel lines are lines in the same plane that never meet. This occurs when they have identical slopes but different y-intercepts. In a linear system, parallel lines indicate that there is no shared solution to the equations. For example, in the system analyzed, both equations transformed to yield the slope \(-\frac{1}{3}\). With varying y-intercepts (0 and -2), the lines are parallel.

• These lines will always maintain a constant distance from each other.
• They provide no point of intersection.

This absence of intersection means the system has no solutions, or it is classified as inconsistent.

Solutions of Linear Systems

The solution to a linear system can be visualized through graphing. When graphing two linear equations, the possible outcomes are:

• One solution: The lines intersect at a single point. This point satisfies both equations, meaning the system is consistent and independent.
• No solution: The lines are parallel and never meet. This results in an inconsistent system, as seen in the given exercise.
• Infinitely many solutions: The lines are coincident (they lie exactly on top of each other), making the system consistent and dependent.

Graphical analysis gives a visual method to understand these relationships, illustrating whether two lines in a linear system converge to a single point, run parallel, or overlap.