Introduction:

When two points lie on the same line, they are said to be collinear. In this case, we are given three points - (x, y), (1, 2), and (7, 0), and we need to determine the relationship between x and y if these points are collinear. To solve this problem, we will use the concept of slope.

Understanding the concept of slope:

The slope of a line is the measure of how steep the line is. It is defined as the change in y divided by the change in x between two points on the line. Mathematically, the slope (m) is given by the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of two points on the line.

Determining the slope:

To determine the slope of the line passing through the points (1, 2) and (7, 0), we can use the formula mentioned above. Let's substitute the coordinates into the formula:

m = (0 - 2) / (7 - 1)
m = -2 / 6
m = -1/3

Using the slope to find the relationship between x and y:

Since the points (x, y), (1, 2), and (7, 0) are collinear, they lie on the same line with the same slope. Therefore, the slope between the points (x, y) and (1, 2) should also be -1/3.

Using the slope formula, we have:

-1/3 = (y - 2) / (x - 1)

Simplifying the equation:

To find the relationship between x and y, we can simplify the equation further. Let's multiply both sides of the equation by (x - 1) to eliminate the denominator:

-1/3 * (x - 1) = y - 2

Multiplying both sides of the equation by -3 to remove the fraction:

x - 1 = -3y + 6

Final equation:

Rearranging the equation, we get:

x + 3y = 7

Therefore, the relationship between x and y if the points (x, y), (1, 2), and (7, 0) are collinear is given by the equation x + 3y = 7. This equation represents the line that passes through these three collinear points.