Scatter plot correlation coefficient

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kidsasd987
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how can I interpret the condition "if sum of (xi-x_bar)^2(yj-y_bar)^2=0 when i=/=j?"

why does this make the line linear?
 
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Since each term is squared, it is positive definite. So for the entire sum to be equal to 0, each individual term has to be 0. Does that help?