How do you solve the simultaneous equations #4x + 6y = 16# and #x + 2y = 5#?

You can try multiplying the second equation by #-3# and then add the two equations ogether (in column):
#{4x+6y=16#
#{color(red)(-3x-6y=-15# add:
#x+0=1#
#x=1#
Substitute into the first:
#4*1+6y=16#
#6y=12#
#y=12/6=2#

Your two equations look like this

#{(4x + 6y = 16), (x + 2y = 5) :}#

The first thing you need to do is use of the equations to solve for one of the two variables, #x# or #y#, then use this value into the other equation.

For simplicity, use the second equation to solve for #x# by adding #-2y# to both sides of the equation

#x + color(red)(cancel(color(black)(2y))) - color(red)(cancel(color(black)(2y))) = 5 - 2y#

#x = 5 - 2y#

Now plug this value for #x# in your first equation and solve for #y#

# 4 * (5 - 2y) + 6y = 16#

#20 - 8y + 6y = 16#

#20 - 2y = 16#

Isolate #y# on one side of the equation

#-2y + color(red)(cancel(color(black)(20))) - color(red)(cancel(color(black)(20))) = 16 - 20#

#-2y = -4 => y = (-4)/(-2) = 2#

Now take this value and use it to determine #x#

#x = 5 - 2y#

#x = 5 - 2 * 2 = 1#

The two solutions to your system are

#{(x = color(green)(1)), (y=color(green)(2)) :}#