Problem 71 - 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. \(L_{1}: 2 x-y=1 ; L_{2}: x+2 y=-1\)

Step-by-Step Solution

Answer

Based on comparison of their slopes, the lines \(L_{1}: 2x - y = 1\) and \(L_{2}: x + 2y = -1\) are neither parallel nor perpendicular.

Step 1: Express the given equations in slope-intercept form (y = mx + b)

To express \(L_{1}: 2x - y = 1\) in the form \(y = mx + b\), you rearrange the equation to find \(y = 2x - 1\). Following the similar procedure for \(L_{2}\), \(x + 2y = -1\) becomes \(y = -0.5x - 0.5\). The coefficient of \(x\) in each equation represents the slope of the line.

Step 2: Compare the slopes

Looking at the equations derived in step 1, the slope \(m_{1}\) of \(L_{1}\) is 2, and the slope \(m_{2}\) of \(L_{2}\) is -0.5. Now, \(m_{1}\) is not equal to \(m_{2}\), so the lines are not parallel. Also, \(m_{1}\) is not equal to the negative reciprocal of \(m_{2}\), so the lines are not perpendicular.

Step 3: Conclusion

So, from step 2, it can be concluded that the given lines \(L_{1}\) and \(L_{2}\) are neither parallel nor perpendicular. This conclusion is based on the comparison of the slopes.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when analyzing relationships between lines. The slope-intercept form of a line's equation is typically expressed as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) designates the y-intercept, the point at which the line crosses the y-axis.

The slope is a measure of how steep the line is, and the y-intercept gives us a fixed point to help graph the line. By converting the standard form equations \( ax + by = c \) into the slope-intercept form, it's easier to identify and understand these characteristics. For instance, from the equation \( 2x - y = 1 \) we can derive the slope-intercept form \( y = 2x - 1 \), where the slope, \( m \), is 2, and the y-intercept, \( b \), is -1.

Equations in slope-intercept form make it straightforward to graph lines on a coordinate plane and to compare their slopes and y-intercepts, which is particularly useful when determining the relationship between multiple lines.

Parallel and Perpendicular Lines

When studying the characteristics of lines, it is essential to recognize when lines are parallel or perpendicular to each other. Parallel lines never meet and have the same slope. This is why comparing the slopes of two lines can determine if they're parallel. If their slopes are equal, the lines are parallel. On the other hand, perpendicular lines intersect at a right angle (90 degrees). The slope of one line is the negative reciprocal of the other if they are perpendicular.

To visualize this, imagine two lines on a graph: if line \( L_{1} \) has a slope \( m \) and line \( L_{2} \) has a slope \( -\frac{1}{m} \), then \( L_{1} \) and \( L_{2} \) are perpendicular. An easy way to remember this is that perpendicular slopes multiply to -1. For instance, if one slope is 2 (as in \( L_{1}: y = 2x - 1 \)), a line perpendicular to it will have a slope of -0.5, which is the negative reciprocal (\( m = -\frac{1}{2} \)).

Slope Comparison

Slope comparison is a tool for determining the relationship between two lines. Once lines have been rewritten in slope-intercept form, their slopes can be easily compared by looking at the coefficient of \( x \). If the slopes are equal, the lines are parallel; if one slope is the negative reciprocal of the other, the lines are perpendicular.

In the given example, line \( L_{1} \) has a slope of 2, and \( L_{2} \) has a slope of -0.5. Since 2 is not equal to -0.5, we can conclude the lines are not parallel. Moreover, because 2 is not the negative reciprocal of -0.5 (which would be -2), the lines are not perpendicular either. Thus, slope comparison has quickly shown us that \( L_{1} \) and \( L_{2} \) are neither parallel nor perpendicular without needing to graph the lines or calculate the angle between them, showcasing the power of algebra in geometry.