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The image of the point with coordinates $(-3,-1)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(5,3)$. Find $m+b$.
 Apr 29, 2023
 #1
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The midpoint of the segment connecting the two points is the point on the line of reflection. This point has coordinates ((−3+5)/2​,(−1+3)/2​)=(1,1). Therefore, the equation of the line of reflection is y=x+1. We can then plug in one of the points to find m+b. Since (5,3) lies on the line, we have 3=5+1 ⇒ m+b=4​.

 Apr 29, 2023
 #2
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Find m + b

 

Hello Guest!

 

\(P_1(-3,1)\\ P_2(5,3)\\ {\color{blue}P(1,2)}\ middle\ of\ the\ route\\ m_{1,2}=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{3-1}{5-(-3)}\\ m_{1,2}=\dfrac{1}{4}\\ m=-\dfrac{1}{m_{1,2}}=-\dfrac{1}{\frac{1}{4}}\\ \color{blue}m=-4\)

 

Point-direction equation of straight lines

\(y=m(x-x_P)+y_P\\ y=-4(x-1)+2\\ y=-4x+4+2\\ \color{blue}y=-4x+6\\ \color{blue}m=-4,\ b=6\\ \color{blue}m+b=2\)

 

laugh  !

 Apr 29, 2023
edited by asinus  Apr 29, 2023
edited by asinus  Apr 29, 2023

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