Problem 82 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that each slope appears visually correct. Use the slopes of the lines to verify your results. (a) \(y=\frac{2}{3} x\) (b) \(y=-\frac{3}{2} x\) (c) \(y=\frac{2}{3} x+2\)

Step-by-Step Solution

Answer

The lines (a) \(y=\frac{2}{3} x\) and (b) \(y=-\frac{3}{2} x\) are perpendicular, while lines (a) \(y=\frac{2}{3} x\) and (c) \(y=\frac{2}{3} x+2\) are parallel.

Step 1: Relationship Identification

Lets first understand the relationship between the lines. The lines (a) \(y=\frac{2}{3} x\) and (b) \(y=-\frac{3}{2} x\) are inversely proportional; the slope of the line (b) is the negative reciprocal of the slope of line (a). This means they are perpendicular. The lines (a) \(y=\frac{2}{3} x\) and (c) \(y=\frac{2}{3} x+2\) have the same slope which means they are parallel.

Step 2: Graph the Equations

To achieve a clear view of the lines, use a graphing utility to graph the three equations in the same viewing window. Make sure to adjust the window so that each slope appears visually correct.

Step 3: Verify Using Slopes

Now verify these relationships by looking at their slopes. The slope of line (a) \(y=\frac{2}{3} x\) is \(\frac{2}{3}\); the slope of line (b) \(y=-\frac{3}{2} x\) is \(-\frac{3}{2}\), which is the negative reciprocal of the slope of line (a), confirming they are perpendicular. The slope of line (c) \(y=\frac{2}{3} x+2\) is \(\frac{2}{3}\), which is the same as that of line (a), confirming they are parallel.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial when analyzing relationships between different lines on a graph. The slope is a measure of how steep a line is, and it's calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. In mathematical terms, if you have two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) is calculated as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

For example, the line \( y = \frac{2}{3}x \) has a slope of \( \frac{2}{3} \), indicating that for every three units the line moves horizontally to the right, it moves up by two units. The slope is positive, reflecting an upward trend as you move from left to right on the graph. Recognizing the slope helps to determine how lines relate to each other—whether they are increasing or decreasing, and at what rate.

Perpendicular Lines

Perpendicular lines are lines that intersect at a 90-degree angle. The key to identifying perpendicular lines is to look at their slopes. If two lines are perpendicular, the slope of one line is the negative reciprocal of the slope of the other. This means that if the slope of the first line is \( a \), the slope of the line perpendicular to it would be \( -\frac{1}{a} \).

In our exercise, line (a) \( y = \frac{2}{3}x \) and line (b) \( y = -\frac{3}{2}x \) are perpendicular. The slope of line (a) \( \frac{2}{3} \) is the negative reciprocal of the slope of line (b) \( -\frac{3}{2} \), confirming this relationship. When graphed, these lines would cross each other, forming right angles.

Parallel Lines

Parallel lines have the distinctive feature of never intersecting, no matter how far they are extended. This is because they have the same slope. Recognizing parallel lines is straightforward: compare their slopes. If the slopes are equal, the lines are parallel. This attribute of parallel lines holds true in any orientation.

Consider lines (a) \( y = \frac{2}{3}x \) and (c) \( y = \frac{2}{3}x+2 \) from the exercise. Since both lines have a slope of \( \frac{2}{3} \), they will never cross each other and are therefore parallel. This demonstrates that while the y-intercept of the line may change (as seen by the '+2' in line (c)), it does not affect the parallel nature as long as the slope remains consistent.

Graphing Calculator

A graphing calculator is an indispensable tool for students studying algebra and other areas of mathematics. It allows for the visualization of equations by plotting them on a coordinate grid. When graphing linear equations, you can input the equation and the calculator automatically provides the graph, showing the line it represents. Moreover, graphing calculators can adjust the viewing window to ensure that the slope of the line appears visually correct, which is particularly important when comparing multiple lines like in our exercise.

Utilizing this tool enhances understanding and simplifies verification of relationships between lines. If lines are parallel, a graphing calculator shows them never crossing, and if they're perpendicular, it clearly displays them intersecting at a right angle. Adjusting the window or 'zooming' in or out aids in accurately portraying the slopes, especially when they are steep or close to being horizontal or vertical.