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A regression line is a straight line that best fits the given set of data points in such a way that it minimizes the sum of the squared distances between the given data points and the fitted line. This line is represented by the linear equation \(y = mx+c\), where 'm' is the slope of the line and 'c' is the y-intercept.

If all but one of the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) lie on a straight line, then we can determine the equation representing this line using any two distinct points on the line, let's say \((x_1, y_1)\) and \((x_2, y_2)\). The slope of the line, denoted as 'm', is calculated as follows: \[m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\] Now, we can find the y-intercept 'c' by plugging one of these points into the general linear equation \(y = mx + c\). Let's use the point \((x_1, y_1)\): \[c = y_{1} - mx_{1}\] Thus, the equation of the line that passes through all but one point can be written as: \[y = mx + c\]

In typical problems, the regression line may or may not pass through the given data points. However, in this case, since all but one of the given points lie on a straight line, the regression line should, at least to some extent, follow the same pattern. Also, since the regression line seeks to minimize the deviation between the given data points and itself, it would try to align closely to this line as well. Since the regression line seeks to minimize the sum of the squared distances between the given data points and the fitted line, it will be pulled towards the outlier point as well. This means that the regression line may not pass exactly through the other points on the straight line. There will be small deviations from the line. However, it will still pass close to all but one of these points. In conclusion, the regression line may not pass exactly through all but one of the points, but it will pass close to them.