In a test (of math in arabic language) we we're asked to find the points of intersection of a circle and a line. Their equation is given.
In the test I solved system of equations made of their equation and in the process I explain my line of thought using the words therefore, then, so, ... etc (in arabic) but in describing that process I acknowledge that the structure of my proof is done by equivalence, that can be easily seen from the context, we're solving an equation so we proceed by equivalence.
But my professor said that it was all wrong and that I should have used the statement is equivalent (in arabic) in each part of my proof and I got all of my exercises wrong, and there were some exercises in the test where in the process we had to solve an equation so he also said that I was wrong because of that reason so I only got 4 out of 20.
Is the professor wrong or I am?
My computer crashed so i couldn't edit so i made a new question. I've added the equations and system of equations with my solutions
Exercise 1
Circle has equation $x^2+y^2+(m+2)x-2my+m^2-36=0$, find center, radius
I found it to be center $(-\frac{m}{2}-1,m)$ radius r= $\sqrt{\frac{m^2}{4}+m+37}$
Exercise 2
Circle has equation $x^2+y^2+2 x - 2y - 4 = 0$ line has equation $x+y-1=0$ find points of intersection
i found $\left(\frac{-1+\sqrt{11}}{2},\frac{3-\sqrt{11}}{2}\right)$ and $\left(\frac{-1-\sqrt{11}}{2},\frac{3+\sqrt{11}}{2}\right)$
for exercise 3 and 4 it is the same and i checked my calculations again and again with the calculator there's no error but yet all of that was marked with 0 for not saying "is equivalent", i didn't use the symbol $\Rightarrow$ All i got was 4 from exercise 5 which had calculation of dot product and of area of triangle where we are given coordinates of two vectors in the plane
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