First, scale the equations such that one of the variables for both equations will have the same coefficient (the sign need not be the same).
#[1] 5m + 3n = 1.5#
#[2] -8m - 2n = 20#
For the above equations, let us multiply #[1]# by #2# and #[2]# by #3#
#[1] => 2(5m + 3n = 1.5)#
#[1] => 10m + 6n = 3#
#[2] => 3(-8m -2n = 20)#
#[2] => -24m -6n = 60#
Now, let's add equations #[1]# and #[2]#
#[1] => 10m + 6n = 3#
#[2] => -24m - 6n = 60#
#[1] + [2] => -14m = 60#
#=> m = -60/14#
#=> m = -30/7#
To get #n#, simply substitute the value of #m# to either #[1]# or #[2]# and solve for n
#[1] => 5m + 3n = 1.5#
#[1] => 10(-30/7) + 6n = 3#
#[1] => -300/7 + 6n = 3#
#[1] => 6n = 3 + 300/7#
#[1] => 6n = (21 + 300) /7#
#[1] => 6n = 321 / 7#
#[1] => n = 321 /(7 * 6)#
#[1] => n = (107 * 3) / (7 * 2 * 3)#
#[1] => n = 107 / 14#
