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