There are two lines in the x-y plane. The points A(-2,5) and B(3,2) lie on line one (L1), C(-1,-2) and D(4,1) lie on line two (L2). Find whether the two lines intersect and the coordinates of the intersection if they do.

We first want to find the equations of the lines. The general equation of a line is: (y-y₁)=m(x-x₁) where (x₁,y₁) are the coordinates of a point on the line and m is the gradient of the line. 
In case of line one, L₁: (y-y₁)=m₁(x-x₁). Let (x₁,y₁) = A(-2,5) and the gradient of the line using the given points: A(x_a,y_a), B(x_b,y_b) => m₁=(y_a-y_b) / (x_a-x_b)= (5-2)/(-2-3)= -3/5             
So we get L₁: (y-5) = -3/5(x+2)
                         y-5 = (-3/5)x - 6/5      /+5
                            y = (-3/5)x + 19/5
Line two, L₂: (y-y2)=m2(x-x₂). Let (x₂,y₂) = C(-1,-2) and the gradient is similarly m₂ = (-2-1) / (-1-4) = -3/-5 = 3/5
L₂: y+1 = 3/5(x+2)
      y+1 = (3/5)x + 6/5    /-1
          y = (3/5)x + 1/5
If the lines intersect we get a point E(x,y), which satisfies both line equations. (If we draw a quick picture we see that the lines intersect). Using the y coordinate of E we get:
(-3/5)x + 19/5 =(3/5)x + 1/5    / *15
         -9x + 57 = 5x + 3           / +9x - 3
                  54 = 14x               /  :14
                    x = 27/7
Using L₂: y = (1/3)(27/7) + 1/5 = 52/35
So the intersection point is E(27/7, 52/35)
Check if E satisfies the equations and the calculation was correct:
L₁: 52/35 = (-3/5)(27/7) + 19/5 = 52/35
L₂​​​​​​​: 52/35 = (1/3)(27/7) + 1/5 = 52/35