Problem 8 - 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. $$\begin{aligned}x+4 y &=8 \\\3 x+y &=3 \\\2 x+8 y &=8\end{aligned}$$

Step-by-Step Solution

Answer

The given linear system has no solutions, as Line 1 and Line 3 are parallel and therefore do not have a point of intersection with each other. Consequently, there is no single point of intersection between all three lines.

Step 1: Rewrite each equation in slope-intercept form (y = mx + b)

To do this, we need to solve each equation for y: For the first equation, solving for y we have: \[ y = \frac{8}{4} - \frac{1}{4}x = 2 - \frac{1}{4}x \] For the second equation, solving for y we have: \[ y = 3 - 3x \] For the third equation, solving for y we have: \[ y = \frac{8}{8} - \frac{2}{8}x = 1 - \frac{1}{4}x \]

Step 2: Determine the slopes and y-intercepts of each line

Now that each equation is in slope-intercept form, we can identify the slope (m) and y-intercept (b) for each line: For the first equation (y = 2 - \frac{1}{4}x): \(m_1 = -\frac{1}{4}\), \(b_1 = 2\) For the second equation (y = 3 - 3x): \(m_2 = -3\), \(b_2 = 3\) For the third equation (y = 1 - \frac{1}{4}x): \(m_3 = -\frac{1}{4}\), \(b_3 = 1\)

Step 3: Sketch the lines on the xy-plane

Using graph paper or a computer software, graph each equation using its slope and y-intercept: 1. Line 1 - Slope: \(-\frac{1}{4}\), Y-intercept: 2 2. Line 2 - Slope: -3, Y-intercept: 3 3. Line 3 - Slope: \(-\frac{1}{4}\), Y-intercept: 1

Step 4: Analyze the intersections

Observe the positions of the lines and find intersections: 1. Line 1 and Line 2 intersect at a point, so there might be at least one unique solution. 2. Line 1 and Line 3 are parallel because they have the same slope but different y-intercepts. Since they are parallel, there is no single point of intersection between Line 1 and Line 3, and consequently, there is no point of intersection between all three lines.

Step 5: Determine the number of solutions

Since there is no single point of intersection between all three lines, the given linear system has no solutions.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a convenient way to write linear equations. It is given by the equation \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept, which is where the line crosses the y-axis.
This form is particularly useful because it allows us to easily identify the slope and the y-intercept, which are crucial for graphing the line and understanding its behavior.
Knowing the slope-intercept form, you can quickly determine how steep a line is and where it will start on the y-axis, facilitating easy sketching or graphing.

Graphing Linear Equations

Graphing linear equations involves plotting the line represented by the equation on an xy-plane. To graph a line in slope-intercept form:

• Identify the y-intercept \( b \) and plot this point on the y-axis.
• Use the slope \( m \), which is the ratio rise/run, to determine another point. For example, a slope of \( \frac{1}{2} \) means rising 1 unit and running 2 units to the right.
• Draw the line through the plotted points, extending it in both directions.

Graphing is a visual way to see the relationships between different lines, which helps in understanding linear systems.
With graphing, you can easily spot intersections and parallels.

Parallel Lines

Parallel lines are lines in a plane that never meet; they have the same slope but may have different y-intercepts. In the equation \( y = -\frac{1}{4}x + 2 \) and \( y = -\frac{1}{4}x + 1 \), both lines have a slope of \( -\frac{1}{4} \).
Because their slopes are equal, these lines are parallel.
Even though they run in the same direction, their different y-intercepts mean they are offset from one another, never crossing at any point on the plane.
Understanding that parallel lines have the same slope helps when working with linear systems, as it indicates that there will be no solutions if lines are parallel and part of the same system, unless they are identical.

Intersection of Lines

The intersection of lines is the point where two lines cross each other. In linear systems, finding this point can determine solutions.
If two lines intersect, they share exactly one point called the solution of the system.
However, if lines are parallel, as mentioned, they do not intersect, and thus have no solutions in a system unless they are equivalent, in which case there would be infinitely many solutions.
To determine an intersection, you can solve two equations simultaneously to find a common point.
Visualizing intersections on a graph can help you easily identify how many solutions a system might have just by looking at how lines relate to each other on the plane.