Question

which statement is true about the graphs of the two lines $y = -8x - \frac{5}{4}$ and $y = \frac{1}{8}x + \frac{4}{5}$? the lines are perpendicular to each other because $-\frac{5}{4}$ and $\frac{4}{5}$ are opposite reciprocals of each other. the lines are neither parallel nor perpendicular to each other because $-\frac{5}{4}$ and $\frac{4}{5}$ are not opposite reciprocals of each other. the lines are perpendicular to each other because $-8$ and $\frac{1}{8}$ are opposite reciprocals of each other.

Explanation:

Step1: Recall slope-intercept form

The equation of a line in slope - intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For the line \(y=-8x-\frac{5}{4}\), the slope \(m_1=-8\). For the line \(y = \frac{1}{8}x+\frac{4}{5}\), the slope \(m_2=\frac{1}{8}\).

Step2: Check for perpendicular lines

Two lines with slopes \(m_1\) and \(m_2\) are perpendicular if \(m_1\times m_2=- 1\). Let's calculate the product of the slopes: \((-8)\times\frac{1}{8}=-1\). Also, if two numbers are opposite reciprocals, their product is - 1. The reciprocal of \(-8\) is \(-\frac{1}{8}\), and the opposite of \(-\frac{1}{8}\) is \(\frac{1}{8}\), so \(-8\) and \(\frac{1}{8}\) are opposite reciprocals. The y - intercepts (\(-\frac{5}{4}\) and \(\frac{4}{5}\)) do not determine if the lines are perpendicular, the slopes do.

Answer:

The lines are perpendicular to each other because \(-8\) and \(\frac{1}{8}\) are opposite reciprocals of each other.