Help with Straight Line Question

Need help with this question, would appreciate seeing as much working as possible, thanks

A triangle has vertices L(2, 3), M(0, 2) and U(4, -2).


Find the coordinates of the orthocentre, H, of triangle LMU.

Thanks, would appreciate the steps for getting the final answer

Find the line that pases through the point M(0,2) and perpendicular to LU(2,3 4,-2):
Slope(m(LU)) = (-2-3) / (4-2) = -5/2
Slope perpendicular to the line m(LU):
m1 = -1/m(LU) = -1/-5/2 = 2/5
m1= 2/5
comparing to (y=mx+b) passing through the point M(0,2)
2 = 2/5x+b
2 = (2.5)*0 +b =>b = 2
y = (2.5)*x+2 = 5/2x+2 :lineM

Find the line that pases through the point L(2,3) and perpendicular to MU(0,2 4,-2):
Slope(m(MU)) = (-2-2) / (4-0) = -4/4 = -1
Slope perpendicular to th line m(M):
m2 = -1/m(LU) = -1/-1 = 1
m2=1
comparing to (y=mx+b) passing through the point L(2,3)
3 = 1*2+b
3 = 2 +b =>b = 1
y = 1*x+1 = x+1 :lineL

Find the line that pases through the point U(4,-2) and perpendicular to ML(0,2 2,3):
Slope(m(LU)) = (3-2) / (2-0) = 1/2
Slope perpendicular to th line m(M):
m3 = -1/m(LU) = -1/1/2 = -2
m3=-2
comparing to (y=mx+b) passing through the point U(4,-2)
-2 = -2x+b
2 = (-2)*4 +b =>b = 10
y = -2*x+10 = -2x+10 :lineU

Solve the eqM(lineM),eqL(lineL) to get the coordinate of th orthocenter.