Find the equation of the line passing through the points (-1,3) and (3,9). If the line also passes through the point (x, x+2), find the value of x ?

First, let's find the slope of the line passing through the points (-1,3) and (3,9).

Slope (m) = (y2 - y1) / (x2 - x1)
m = (9 - 3) / (3 - (-1))
m = 6 / 4
m = 3/2

Now, we can use the point-slope form of the equation of a line to find the equation of the line passing through the points (-1,3) and (3,9).

y - y1 = m(x - x1)
y - 3 = (3/2)(x - (-1))
y - 3 = (3/2)(x + 1)
y - 3 = (3/2)x + 3/2
y = (3/2)x + 3/2 + 3
y = (3/2)x + 9/2

Now, we know that the line also passes through the point (x, x+2). We can substitute this point into the equation of the line to find the value of x.

x+2 = (3/2)x + 9/2
2 = (3/2)x - x + 9/2
2 = (3/2 - 2)x + 9/2
2 = (3 - 4)x + 9/2
2 = -x + 9/2
2 - 9/2 = -x
4/2 - 9/2 = -x
-5/2 = -x
x = 5/2

Therefore, the value of x is 5/2.

Equation of the line passing through the points (-1,3) and (3,9) is
$$(y-3)/(9-3) = (x-(-1))/(3-(-1))$$
$$(y-3)/6 = (x+1)/4$$
$$4(y-3)=6(x+1)$$
$$4y-12=6x+6$$
$$4y=6x+18$$
$$2y=3x+9$$
$$y=(3/2)x+(9/2)$$

Since the line passes through the point (x, x2), we have
$$x^2=(3/2)x+(9/2)$$
$$x^2-(3/2)x-(9/2)=0$$
$$2x^2-3x-9=0$$
$$x=[3\pm\sqrt{9+4(9)}]/4$$
$$x=[3\pm5]/4$$
$$x=8/4 \ \text{or} \ x=-2/4$$
$$x=2 \ \text{or} \ x=-1/2$$

Hence, the value of x is 2 or -1/2.