Problem 17 - Precalculus: Mathematics for Calculus

Question

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. $$\left\\{\begin{array}{l} 2 x-3 y=12 \\ -x+\frac{3}{2} y=4 \end{array}\right.$$

Step-by-Step Solution

Answer

The system has no solution; the lines are parallel.

Step 1: Convert Equations to Slope-Intercept Form

First, we'll convert both equations into the slope-intercept form, which is \(y = mx + b\). For the first equation, \(2x - 3y = 12\), we solve for \(y\):\[2x - 3y = 12\]\[-3y = -2x + 12\]\[y = \frac{2}{3}x - 4\].For the second equation, \(-x + \frac{3}{2}y = 4\), we solve for \(y\):\[-x + \frac{3}{2}y = 4\]\[\frac{3}{2}y = x + 4\]\[y = \frac{2}{3}x + \frac{8}{3}\].

Step 2: Plot the Lines on Graph

Now that we have both equations in slope-intercept form, plot each line on a graph. For \(y = \frac{2}{3}x - 4\) and \(y = \frac{2}{3}x + \frac{8}{3}\), you will find both have the same slope (\(\frac{2}{3}\)), which means they are parallel.

Step 3: Analyze the Graph

Since the lines are parallel and do not intersect, the system of equations has no solution. Parallel lines with different y-intercepts do not meet at any point on the graph.

Key Concepts

Slope-Intercept Form

The easiest way to understand linear equations is by using the slope-intercept form. This form of a line is given by the equation \(y = mx + b\), where \(m\) represents the slope, and \(b\) stands for the y-intercept. Here is what each component means:

• **Slope \(m\):** This indicates the steepness and direction of the line. It tells us how much \(y\) changes for a one-unit change in \(x\). A positive slope means the line rises as we move to the right, while a negative slope means it falls.
• **Y-Intercept \(b\):** This is the point where the line crosses the y-axis. It shows the value of \(y\) when \(x\) is zero.

Getting your equations into this form makes it much easier to graph them, as it gives you a clear starting point (the \(y\)-intercept) and direction (the slope) for drawing your line. In this exercise, converting the equations was the first step, allowing us to derive the lines' slopes \(\left(\frac{2}{3}\right)\) and y-intercepts \(-4\) and \(\frac{8}{3}\).

Parallel Lines

Once you have equations in slope-intercept form, you can quickly determine if two lines are parallel. Parallel lines are lines in a plane that never meet; they stay the same distance apart no matter how far they are extended. For two lines to be parallel, they must have the same slope.

In the given example, both lines were converted to slope-intercept form as \(y = \frac{2}{3}x - 4\) and \(y = \frac{2}{3}x + \frac{8}{3}\). Notice how both lines share the slope \(\frac{2}{3}\), which means they are parallel. However, the different y-intercepts \(-4\) and \(\frac{8}{3}\) ensure that these lines do not overlap—they run alongside each other but never cross.

No Solution

Understanding when a system of equations has no solution involves looking at the relationship between the lines representing those equations. If lines are parallel and do not intersect, they do not share any points in common. Hence, the system is said to have no solution.

In mathematical terms, when there is no point \((x, y)\) that simultaneously satisfies both equations, the system has no solution. This is evident in our graph, where the lines \(y = \frac{2}{3}x - 4\) and \(y = \frac{2}{3}x + \frac{8}{3}\) do not meet at any point. This confirms that there is no solution to the system of equations because the lines are parallel and distinct.