Question

the equation of (overleftrightarrow{xy}) is (y = -x - 5). (overleftrightarrow{xy}) is the image of (overleftrightarrow{xy}) after a dilation with a scale factor of 3 centered at ((1, -3)). what is the slope of (overleftrightarrow{xy})? write your answer as an integer or simplified fraction.

Explanation:

Step1: Recall dilation property on slope

A dilation is a transformation that scales the figure. For a line, dilation (centered at any point) does not change the slope of the line. Because dilation preserves the direction of the line, so the slope remains the same.

Step2: Identify slope of original line

The original line \(\overleftrightarrow{XY}\) has the equation \(y=-x - 5\). The slope - intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. Comparing \(y=-x - 5\) with \(y = mx + b\), we can see that the slope \(m=- 1\).

Step3: Determine slope of image line

Since dilation does not change the slope of a line, the slope of \(\overleftrightarrow{X'Y'}\) (the image of \(\overleftrightarrow{XY}\) after dilation) is the same as the slope of \(\overleftrightarrow{XY}\).

Answer:

\(-1\)