Problem 5 - 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. $$\begin{aligned}&3 x-4 y=12\\\&6 x-8 y=24\end{aligned}$$

Step-by-Step Solution

Answer

The given linear system can be rewritten as: 1. \(y = \frac{3}{4}x - 3\) 2. \(y = \frac{3}{4}x - 3\) Since both equations have the same slope (\(\frac{3}{4}\)) and the same y-intercept (-3), they represent the same line in the xy-plane. Therefore, the linear system has infinitely many solutions.

Step 1: Convert Equation 1 to Slope-Intercept Form

For the first equation: \(3x - 4y = 12\) 1. First, we need to isolate y. Subtract 3x from both sides of the equation: \(-4y = -3x + 12\) 2. Now, divide both sides of the equation by -4: \(y = \frac{3}{4}x - 3\)

Step 2: Convert Equation 2 to Slope-Intercept Form

For the second equation: \(6x - 8y = 24\) 1. First, we need to isolate y. Subtract 6x from both sides of the equation: \(-8y = -6x + 24\) 2. Now, divide both sides of the equation by -8: \(y = \frac{3}{4}x - 3\)

Step 3: Analyze the Slopes and Y-intercepts

Now we have both equations in slope-intercept form: 1. Equation 1: \(y = \frac{3}{4}x - 3\) 2. Equation 2: \(y = \frac{3}{4}x - 3\) We can see that both equations have the same slope (\(\frac{3}{4}\)) and the same y-intercept (-3).

Step 4: Sketch the Equations and Determine the Number of Solutions

Since both equations have the same slope and y-intercept, they represent the same line in the xy-plane. When we sketch the equations, we'll see that they overlap completely. In this case, the linear system has infinitely many solutions.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express linear equations in an easy-to-read format. It is written as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) represents the y-intercept. The slope \(m\) indicates how steep the line is, and it determines the direction of the line—positive slopes rise to the right, while negative slopes fall to the right.
Understanding this form makes it simpler to analyze a line's behavior with just a glance. In our exercise, both equations from the given system have been converted into this form. We can see that both equations simplify to \(y = \frac{3}{4}x - 3\).

• The slope is \(\frac{3}{4}\), indicating a moderate rise as \(x\) increases.
• The y-intercept is \(-3\), which tells us where the line crosses the y-axis.

This format allows us to quickly compare the two equations by examining their slopes and y-intercepts.

Infinitely Many Solutions

When solving systems of linear equations, it's possible to encounter scenarios where there are infinitely many solutions. This occurs when the two equations you're working with describe the exact same line.
In our problem, after converting the equations to the slope-intercept form, we notice the following characteristics are identical:

• Both lines have a slope of \(\frac{3}{4}\).
• Both lines have a y-intercept of \(-3\).

These identical characteristics mean that the lines overlap perfectly in the xy-plane. Because they coincide completely, every point on one line is also on the other line, thus there are infinitely many points that satisfy both equations. It's crucial to recognize this situation, as it tells us that there is not just a single solution, but rather an infinite number of solutions.

Sketching Solutions

Graphing linear equations provides a visual representation and can simplify the process of understanding whether and how they intersect. The task in our exercise involves sketching both equations to determine the number of solutions visually.
By converting the equations to slope-intercept form, we prepare them for an easier sketching process. Here's how you could visualize it:

• Start by plotting the y-intercept on the graph. For our equations, locate \(y = -3\) on the y-axis.
• Use the slope \(\frac{3}{4}\) to find another point. From the y-intercept, move up 3 units and right 4 units to plot another point.
• Draw the line through these points. Repeat for both equations.

Upon sketching, you'll notice these are not two separate lines, but a single line drawn twice. They completely overlap, reaffirming that there are infinitely many solutions. Sketching offers a valuable perspective that complements algebraic techniques, reinforcing the understanding of linear systems.