Problem 6 - Differential Equations and Linear Algebra

Question

Make a sketch in the \(x y\) -plane in order to determine the number of solutions to the given linear system. $$\begin{aligned}&2 x+3 y=1\\\&2 x+3 y=2\end{aligned}$$

Step-by-Step Solution

Answer

The given linear system consists of two parallel lines with different y-intercepts in the xy-plane: \(y = -\frac{2}{3}x + \frac{1}{3}\) and \(y = -\frac{2}{3}x + \frac{2}{3}\). Since they do not intersect, there are no solutions for the system.

Step 1: Convert the equations to slope-intercept form y = mx + b

To graph the lines on the xy-plane, we need to convert them into the slope-intercept form, which is y = mx +b, where m is the slope and b is the y-intercept. Given equations: 1) \(2x + 3y = 1\) 2) \(2x + 3y = 2\) For equation 1, we can solve it for y: 3y = -2x + 1 y = -\(\frac{2}{3}\)x + \(\frac{1}{3}\) For equation 2, we can also solve it for y: 3y = -2x + 2 y = -\(\frac{2}{3}\)x + \(\frac{2}{3}\) Now, we have the slope-intercept form for both equations: 1) \(y = -\frac{2}{3}x + \frac{1}{3}\) 2) \(y = -\frac{2}{3}x + \frac{2}{3}\)

Step 2: Identify the slopes and y-intercepts of the equations

Now, we can easily identify the slopes (m) and y-intercepts (b) of the two lines: For equation 1: Slope(m1) = -\(\frac{2}{3}\), y-intercept(b1) = \(\frac{1}{3}\) For equation 2: Slope(m2) = -\(\frac{2}{3}\), y-intercept(b2) = \(\frac{2}{3}\)

Step 3: Make a sketch of the two lines in the xy-plane

Now that we have the slope-intercept form for both lines, we can make a sketch in the xy-plane. 1) For equation 1, plot the y-intercept at \(\frac{1}{3}\) on the y-axis, and then use the slope to draw the line with the slope of -\(\frac{2}{3}\). 2) For equation 2, plot the y-intercept at \(\frac{2}{3}\) on the y-axis, and then use the slope to draw the line with the slope of -\(\frac{2}{3}\). Notice that both lines have the same slope, so they are parallel.

Step 4: Analyze the sketch to determine the number of solutions

Since both lines are parallel with different y-intercepts, they will never intersect. Hence, the given linear system has no solutions.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a popular way to express straight lines on a graph and is represented as \( y = mx + b \). Here, \( m \) stands for the slope of the line, indicating its steepness or incline, while \( b \) is the y-intercept, where the line crosses the y-axis. By converting equations into this form, we can easily understand the line's behavior on a graph.
In our example, we have two equations:

• \( 2x + 3y = 1 \)
• \( 2x + 3y = 2 \)

By solving these equations for \( y \), each is transformed into the slope-intercept form:

• For the first equation: \( y = -\frac{2}{3}x + \frac{1}{3} \)
• For the second equation: \( y = -\frac{2}{3}x + \frac{2}{3} \)

This conversion helps in plotting them on a graph, revealing key features like parallelism.

Parallel Lines

Parallel lines are lines in a plane that never intersect. They have the same slope but different y-intercepts. This means they run alongside each other, maintaining a constant distance between them. In the context of our system of equations, both lines derived from the equations \( y = -\frac{2}{3}x + \frac{1}{3} \) and \( y = -\frac{2}{3}x + \frac{2}{3} \) are parallel.

• Both lines have the slope \( -\frac{2}{3} \).
• However, their y-intercepts are different: \( \,\frac{1}{3} \, \) for the first and \( \,\frac{2}{3} \) for the second.

These lines, with identical slopes but distinct y-intercepts, illustrate the concept of parallelism perfectly. No matter how far you extend these lines on the graph, they will never meet.

Number of Solutions

In linear systems, the number of solutions depends on how the lines interact on a graph. There can be three possible scenarios:

• One solution: If the lines intersect at exactly one point.
• Infinite solutions: If the lines overlap completely, indicating they are the same line.
• No solution: If the lines are parallel and never meet.

For our given linear system, since the lines are parallel, as shown by their same slope and different y-intercepts, they have no common point of intersection. Therefore, the system has no solutions. This scenario highlights the importance of analyzing line characteristics in determining the nature of solutions for linear systems.