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Chapter 1: Problem 41

For what value of \(k\) are the two lines \(2 x+k y=3\) and \(x+y=1\) (a) parallel? (b) perpendicular?

Short Answer

Expert verified
For the lines to be parallel, \(k = 2\). For the lines to be perpendicular, \(k = 2\). The result is essentially the same in this special case.

Step by step solution

01

Convert to slope-intercept form

Let's convert both equations into slope-intercept form. \n For \(2x + ky = 3\), we get \(y = (-2/k)x + (3/k)\). So, the slope is \(-2/k\). \n For \(x+y=1\), we get \(y = -x + 1\). So, the slope is -1.
02

Find \(k\) for the lines to be parallel

For the lines to be parallel, their slopes must be equal. Thus, we equate \(-2/k\) and -1 which gives us \(k = -2/k = -1\). Therefore, \(k = 2\). This is the value of \(k\) when the two lines are parallel.
03

Find \(k\) for the lines to be perpendicular

For the lines to be perpendicular, the product of their slopes must be -1. Thus, we find \(k\) such that \(-2/k * -1 = -1\). Solving, we find \(k = 2\). Therefore, \(k = 2\) is the value of k when the two lines are perpendicular.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parallel Lines
Parallel lines are lines in the same plane that never intersect. No matter how far they are extended, parallel lines will always remain the same distance apart. We can easily identify two lines are parallel if they have the same slope. For instance, if line 1 has a slope of \(m_1\) and line 2 has a slope of \(m_2\), for the lines to be parallel, \(m_1 = m_2\).
In the original exercise, we have two equations: \(2x + ky = 3\) and \(x + y = 1\). By using the slope-intercept form \(y = mx + c\), we converted them to find their slopes. The slope of the first line is \(-2/k\) and the slope of the second line is \(-1\). To ensure these lines are parallel, set \(-2/k = -1\). Solving gives us \(k = 2\). Thus, when \(k = 2\), the lines are parallel.
Perpendicular Lines
Perpendicular lines are lines that intersect to form right angles, which means they meet at a 90-degree angle. For two lines to be perpendicular, the product of their slopes must be \(-1\). In mathematical terms, if the slope of line 1 is \(m_1\) and the slope of line 2 is \(m_2\), then \(m_1 \times m_2 = -1\).
In this exercise, we use the same converted equations where the slopes are \(-2/k\) and \(-1\). To find when these lines are perpendicular, we set \(-2/k \times -1 = -1\). Solving this equation results in \(k = 2\). So, \(k = 2\) ensures the lines are perpendicular and intersect at right angles.
Slope-Intercept Form
The slope-intercept form of a linear equation enables us to easily identify important features of a line. It is expressed as \(y = mx + c\), where \(m\) represents the slope of the line and \(c\) denotes the y-intercept. This form is crucial for quickly determining how a line behaves on a graph. The slope \(m\) shows the rise over run, or how steep a line is. The y-intercept \(c\) is where the line crosses the y-axis.
In our problem, converting equations to the slope-intercept form made it straightforward to analyze the relationships between the lines. The equation \(2x + ky = 3\) transformed to \(y = (-2/k)x + 3/k\) gives a slope of \(-2/k\), and \(x + y = 1\) transformed to \(y = -x + 1\) shows a slope of \(-1\). Having the equations in this form allowed us to easily deduce conditions for parallelism and perpendicularity between the lines by comparing their slopes.

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Most popular questions from this chapter

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