How do you solve #x − 5 = −10#?

x - 5 = -10
Bringing the constants to one side, and the variable on the other; remember to change the signs as you shift
x = -10 + 5
x = -5

Given: #color(brown)(color(white)(..)x-5=-10)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Demonstration of the concept behind process")#

Consider 6=6

This we know to be true

Suppose we subtracted 2 only from the left hand side (LHS)

#color(red)("Then it would be totally wrong to write "6-2=6)# because the equals sign is stating that the total value on LHS is exactly the same value on the right hand side (RHS).

To make this be correct we apply the golden rule:
#color(green)("What we do to one side of the equals sign we do to the other.")#

so we subtract 2 #color(purple)("from both sides")#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Objective")#
End up with #x# on its own on the LHS of = and everything else on the other

#color(blue)("To 'remove' the -5 from LHS")#

Add #color(blue)(5)# to both sides of the equation

#color(brown)((x-5)color(blue)(+5) =(-10)color(blue)(+5) #
The brackets are only there to show the original parts of the equation.

#x+5-5=5-10#

But +5-5=0 and 5-10 = -5 giving

#x+0=-5#

#x=-5#

The shortcut gets you to the same place but very much quicker.
With addition; to move something to the other side of the equals sign do so but change its sign.

By shortcut:
#x-5=-10 #
#x=-10+5#
#x=-5#