Problem 74 - College Algebra and Calculus: An Applied Approach

Question

The equations of two lines are given. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. \(L_{1}: 4 x-y=-2 ; L_{2}: 8 x-2 y=6\)

Step-by-Step Solution

Answer

Lines \(L_{1}\) and \(L_{2}\) are parallel since their slopes are equal.

Step 1: Transform the Equations to Slope-Intercept Form

First, let's transform the equations into slope-intercept form (y=mx+b). For the equation \(L_{1}: 4x -y= -2\), we rearrange it to get \(y = 4x + 2\). For the equation \(L_{2}: 8x - 2y = 6\), we rearrange it to get \(y = 4x - 3\).

Step 2: Calculate and Compare the Slopes.

From the transformed equations, we find that the slopes of both lines are the same, \(m_{1} = 4\) and \(m_{2} = 4\). Consequently, since the slopes are equal, we can conclude that the lines are parallel.

Step 3: Check if the Lines are Perpendicular

Since our slopes are not negative reciprocals of each other, it implies that these two lines are not perpendicular.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is fundamental to analyzing line relationships. It allows for quick identification of the slope and y-intercept of a line. The slope-intercept form is expressed as (y = mx + b), where (m) is the slope, and (b) is the y-intercept, the point where the line crosses the y-axis. For a clear understanding, let's convert the given lines into this form.
Starting with the first line (4x - y = -2), we add (y) to both sides and then subtract (2), which gives us the final form (y = 4x + 2). For the second line (8x - 2y = 6), we first divide every term by (2) to isolate y, resulting in (y = 4x - 3). This representation makes it easier to compare the lines in terms of slope and intercept.

Parallel Lines

Parallel lines are lines in the same plane that never intersect; they have the same slope but different y-intercepts. When analyzing equations of lines like those given for (L_{1}) and (L_{2}), we see that after converting to slope-intercept form, both lines have the same slope of (4). This is a clear indicator that the lines are parallel. To visualize, you can think of railroad tracks that remain at a consistent distance from each other and thus never meet. This concept is vital when interpreting graphs and can also be used to solve problems involving geometric figures formed with parallel lines and when designing structures requiring parallel supports.

Perpendicular Lines

Conversely, perpendicular lines intersect at a right angle (90 degrees). The key to identifying perpendicular lines is to look at the slopes: they are negative reciprocals of each other. This means if one line has a slope of (m), the other must have a slope of (-1/m). In our example, since both lines have a slope of (4), and not one having (-1/4), they cannot be perpendicular. When two lines are perpendicular, their slopes multiply together to equal (-1). This concept is not only crucial in geometry, but also in fields like architecture and engineering where right angles are frequently employed in design and construction.

Comparing Slopes

Comparing slopes is a straightforward way to determine the relationship between two lines. If their slopes are identical, as seen with slopes (m_{1} = 4) and (m_{2} = 4), the lines are parallel. If the product of their slopes equals (-1), indicating they are negative reciprocals of each other, the lines are perpendicular. In cases where the slopes are neither identical nor negative reciprocals, the lines are simply neither parallel nor perpendicular. Understanding slope comparisons is essential in coordinate geometry, optimization problems, and in real-life situations such as calculating the inclination of a ramp or understanding the trajectory of paths that may cross or run alongside each other.