Problem 47 - Contemporary Precalculus

Question

Determine whether the lines whose equations are given are parallel, perpendicular, or neither. \(2 x+y-2=0 \quad\) and \(\quad 4 x+2 y+18=0\)

Step-by-Step Solution

Answer

Answer: The lines are parallel.

Step 1: Rewrite the given equations in slope-intercept form

For the first line, \(2x + y - 2 = 0\), we can rewrite it as: \(y = -2x + 2\) For the second line, \(4x + 2y + 18 = 0\), we can rewrite it as: \(2y = -4x - 18\) \(y = -2x - 9\) Now we have both lines in slope-intercept form.

Step 2: Identify the slopes of the lines

For the first line, \(y = -2x + 2\), we see that the slope (m1) is -2. For the second line, \(y = -2x - 9\), we see that the slope (m2) is also -2.

Step 3: Compare the slopes

Now that we have the slopes of both lines, we can compare them to determine the relationship between the lines. Since both slopes are equal (m1 = m2 = -2), the lines with the given equations are parallel.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express a linear equation. It is written as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept. The y-intercept is the point where the line crosses the y-axis.

Converting an equation into the slope-intercept form is quite straightforward. You need to manipulate the equation to isolate \(y\) on one side of the equation. For example, let's talk about the line \(2x + y - 2 = 0\):

• Subtract \(2x\) from both sides to get \(y - 2 = -2x\).
• Then, add \(2\) to both sides to find \(y = -2x + 2\).

Now, we can clearly see the slope \(m = -2\) and the y-intercept \(b = 2\).

This form allows for easy identification of the slope and y-intercept, enabling analysis of the line's behavior and its intersections.

Line Equations

Line equations can come in various forms, but among the most useful when comparing lines are the standard form and the slope-intercept form. In this article, we're focusing on the slope-intercept form due to its simplicity and ease in revealing the slope and y-intercept directly.

Let's consider another line \(4x + 2y + 18 = 0\). To write it in slope-intercept form, follow these steps:

• First, move \(4x\) and \(18\) to the other side to get \(2y = -4x - 18\).
• Divide each term by \(2\) to isolate \(y\), resulting in \(y = -2x - 9\).

With this, we get the slope as \(-2\) and see that it intersects the y-axis at \(-9\).

The clarity provided by this form is what makes it extremely useful in geometry when comparing different lines and their characteristics.

Comparing Slopes

When it comes to comparing lines, analyzing their slopes is key. Slopes indicate how steep a line is and in which direction it moves. In our examples, both lines were transformed into the slope-intercept form:

• Line 1: \(y = -2x + 2\) with a slope of \(-2\).
• Line 2: \(y = -2x - 9\) with the same slope \(-2\).

Since both lines have identical slopes, we can conclude that they are parallel. Lines that are parallel never intersect and move in the same direction at the same angle.

It's also helpful to know that if two lines were to be perpendicular, their slopes would be negative reciprocals of each other. For instance, a line with a slope of \(-2\) would be perpendicular to a line with a slope of \(\frac{1}{2}\).

Understanding how to compare slopes gives you the ability to determine the relationship between multiple lines quickly and accurately.