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Chapter 2: Problem 7

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((2,-3)\) and perpendicular to the line whose equation is \(y=\frac{1}{5} x+6\)

Short Answer

Expert verified
The equation of the line that is perpendicular to \(y=\frac{1}{5}x + 6\) and passes through the point (2,-3) is \(y + 3 = -5x + 10\) in point-slope form, and \(y = -5x + 7\) in slope-intercept form.

Step by step solution

01

Find the slope of the given line

Analyze the equation of the given line, \(y=\frac{1}{5}x+6\), in its slope-intercept form, \(y=mx+b\), where m indicates the slope. So in this case, slope \((m_1)=\frac{1}{5}\).
02

Determine slope of the perpendicular line

Lines are perpendicular if the product of their slopes is -1. This property helps to determine the slope \((m_2)\) of the line perpendicular to the given line, by using the formula \(m_1*m_2 = -1\). Substituting \(m_1 = \frac{1}{5}\), the formula changes to \(\frac{1}{5} * m_2 = -1\), which can be solved to \(m_2 = -5\).
03

Write the equation in point-slope form

Using the point-slope formula for the equation of a line, \(y - y_1 = m(x - x_1)\), where \(x_1=2\) and \(y_1=-3\) are the coordinates of a point on the line and \(m=-5\) is the slope of the line. Substituting these values in, the equation becomes \(y - (-3) = -5*(x - 2)\), which simplifies to \(y + 3 = -5x + 10\).
04

Convert equation into slope-intercept form

To transform the equation into the slope-intercept form \(y = mx + b\), it's needed to isolate y. In order to achieve that, -3 is subtracted from both sides of the equation, which results in \(y = -5x + 7\) .

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