The sides of a triangle are given by the equations 4x-y-7=0,x+3y-31=0 and x+5y-7=0.determine the point of intersection of its altitudes
details: class 12,algebra,
title of chapter:
the general equations of straight lines.

Hello,

1)Let's call A=inter (4x-y-7=0, x+3y-31=0)
4x-y=7
x+3y=31
==>13x=52==>x=4
and y=4*4-7=9

A=(4;9)

 
2)Let's call B=inter (4x-y-7=0, x+5y-7=0)

4x-y=7
x+5y=7
==>21x=42
==>x=2 and y=4*2-7=1

B=(2,1)

3) Slope of BD: y=-x/3+31/3 ==> slope perpendicular=3

Line BD:y-1=3(x-2)==>y=3x+5

4) Slope AE: y=-x/5+7/5 ==>slope perpendicular=5
Line AE: y-9=5(x-4)==> y=5x-11

5) intersection (BD;AE):
y=5x-11
y=3x-5
==>2x=6==>x=3
and y=3*3-5=4
H=(3,4)