Problem 65 - Calculus Single Variable

Question

Suppose that \(A\) and \(B\) are constants that are not both zero and that \(D\) and \(E\) are any two constants. Prove that the lines \(A x+B y=D\) and \(-B x+A y=E\) are perpendicular.

Step-by-Step Solution

Answer

The lines are perpendicular because the product of their slopes is \(-1\).

Step 1: Identify slope-intercept form

Convert the general line equations into slope-intercept form \(y = mx + c\) to identify their slopes. The first equation is \(A x + B y = D\); rearrange this to get \(B y = -A x + D\), then \(y = \frac{-A}{B}x + \frac{D}{B}\). Thus, the slope \(m_1\) of the first line is \(\frac{-A}{B}\).

Step 2: Determine slope of the second line

Convert the second equation \(-B x + A y = E\) into slope-intercept form. Rearrange to get \(A y = B x + E\), then \(y = \frac{B}{A}x + \frac{E}{A}\). Therefore, the slope \(m_2\) of the second line is \(\frac{B}{A}\).

Step 3: Check perpendicularity condition

Two lines are perpendicular if the product of their slopes is \(-1\). Calculate the product of the slopes: \(m_1 \times m_2 = \left(\frac{-A}{B}\right) \times \left(\frac{B}{A}\right) = \frac{-AB}{AB} = -1\). Since the product is \(-1\), the lines are perpendicular.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial for recognizing and analyzing lines on a graph. The slope-intercept form is expressed as \(y = mx + c\), where \(m\) represents the slope of the line, and \(c\) signifies the y-intercept, which is the point where the line crosses the y-axis.
By converting linear equations into this form, you can easily visualize their behaviors on a graph. For instance, if you have a line with equation \(A x + B y = D\), rearranging it to fit into the slope-intercept form involves isolating \(y\) on one side. So, you would get \(y = \frac{-A}{B}x + \frac{D}{B}\).
In practical problems like the one above, identifying the slope-intercept form can help determine relationships between lines, such as parallelism or perpendicularly, by focusing on their slopes.

Linear Equations

A linear equation is an algebraic expression that represents a straight line when plotted on a graph. It typically looks like \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. The key characteristic of a linear equation is its constant slope and straight-line graph result.
When working with linear equations, it's often useful to rearrange them into the slope-intercept form, as this allows for easy identification of the slope and y-intercept, providing insight into the line's direction and position on a graph.
In addition to slope, the concept of linear equations involves understanding how constants shift the line vertically or horizontally, effectively changing the graph's crossing points on respective axes.

Slopes of Lines

The slope of a line is a measure of its steepness and direction, calculated as the change in y over the change in x, often expressed as \(m = \frac{\Delta y}{\Delta x}\). A line's slope determines whether it inclines upwards, declines downwards, or remains flat.
When considering two lines, the slopes can reveal their specific relationship in terms of parallelism or perpendicularity. If two lines have the same slope, this means they run parallel. Conversely, if two lines have slopes that multiply to -1, they are perpendicular. This is because perpendicular lines intersect at a right angle, effectively reversing and inverting steepness.
Understanding how to compute and compare slopes is essential in geometry and applied mathematics, providing clarity on how different linear equations interact.