Determine m given that m>0 and the points (m,3) and (1,m) lie on a line with slope m.

To find the value of m in this problem, we can use the concept of the slope of a line. The slope is a measure of how steep the line is and can be determined using the formula `(y2 - y1) / (x2 - x1)`, where `(x1, y1)` and `(x2, y2)` are two points on the line.

In this case, we have two points on the line: (m, 3) and (1, m). We are also given that the slope of the line is m itself. So we can set up the equation:

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

Plugging in the coordinates, we get:

m = (m - 3) / (1 - m)

To solve for m, we can cross-multiply:

m(1 - m) = m - 3

Expanding the brackets:

m - m^2 = m - 3

Rearranging the equation:

m^2 - m - 3 = 0

Now we have a quadratic equation. We can solve it by factoring or using the quadratic formula. By factoring, we would need to find two numbers that multiply to -3 and add up to -1. However, this quadratic equation does not factor nicely.

So, let's use the quadratic formula, which states that for an equation `ax^2 + bx + c = 0`, the solution for x is given by:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

Applying this to our equation:

m = (-(-1) ± sqrt((-1)^2 - 4*1*(-3))) / (2*1)

Simplifying:

m = (1 ± sqrt(1 + 12)) / 2

m = (1 ± sqrt(13)) / 2

Therefore, the solutions for m are:

m = (1 + sqrt(13)) / 2 or m = (1 - sqrt(13)) / 2

However, we are given that m > 0, so the only valid solution in this case is:

m = (1 + sqrt(13)) / 2