The line passing through the points (1, 1) and (5, 5) can be determined by finding its slope and using the point-slope form of a line.
First, we calculate the slope (m) of the line:
m = (y2 – y1) / (x2 – x1) = (5 – 1) / (5 – 1) = 4 / 4 = 1.
Since the slope is 1, we can use the point-slope form of the equation: y – y1 = m(x – x1). Using point (1, 1):
y – 1 = 1(x – 1)
This simplifies to:
y = x.
Another point we could use is (5, 5):
y – 5 = 1(x – 5)
This simplifies to:
y = x.
Thus, any equation of the line can be represented as y = x + b, where b is a constant that makes the equation pass through the point (1, 1) or (5, 5). Examples of correct equations would be y = x, or y – x = 0.
Now, an example of an equation that does not represent this line could be:
y = 2x.
This is not correct because it has a slope of 2, while our line has a slope of 1. Therefore, it does not pass through (1, 1) or (5, 5).