Coordinates of line points

I have a point A=[2,1] and a line y+x-5=0. How do I determine the coordinates of point A', which is the image of point A according to the given straight line?

Correct answer:

x2 =  4
y2 =  3

Step-by-step explanation:

A=(2,1) A′=(x2​,y2​) p: x+y−5=0  q ⊥ p; A ∈ q q:x−y+c=0  (1,1)⋅ (1,−1) = 0  Ax​−Ay​+c=0 c=Ay​−Ax​=1−2=−1  q: x−y−1=0  S = p ∩ q  x0​+y0​−5=0 x0​−y0​−1=0  x0​+y0​=5 x0​−y0​=1  Row2−Row1→Row2 x0​+y0​=5 −2y0​=−4  y0​=−2−4​=2 x0​=5−y0​=5−2=3  x0​=3 y0​=2  S=(x0​,y0​) S=(3,2)  x2​=−Ax​+2⋅ Sx​=−2+2⋅ 3=4
y2​=−Ay​+2⋅ Sy​=−1+2⋅ 2=3

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