How do you solve the following system: #x - 3y = -1 , 8x-y=19 #?

The objective when solving an equation is to have only 1 unknown and some numeric values. This is then solvable.

For a system to be solvable you have to have the same count of equations as the count of unknowns.

You have two unknowns and two equations so this system is solvable.
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#color(blue)("Initial approach")#

Change one of the equations so that there is the same count of the variable #y# in each of them. Then by subtraction remove this variable. Hence solve for #x#

#color(blue)("Solving for "x)#

Given:
#" "x-3y=-1" "#..................................(1)
#" "8x-y=19" "#.....................................(2)

Multiply equation (2) by 3 so that we have the same number of #y's# in each.

#" "x-3y=-1" "#..................................(1)
#" "underline(24x-3y=57" ")..........................(2_a)#

I do not like it this way round for subtraction so reverse order

#" "24x-3y=57" "..........................(2_a)#
#" "underline(color(white)(..)x-3y=-1" ")#..................................(1)
Subtraction#" "23x +0color(white)(.)=58#

Divide both sides by 23 giving

#" "color(blue)(x=58/23)#
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#color(blue)("Solving for "y)#

Substitute #color(blue)(x=58/23)# into equation (1)
I chose equation (1) to make the calculation easier!

#color(brown)(x-3y=-1" "->" "color(blue)((58/23))-3y=-1)#

#" "3y=58/23+1#

#y=58/(23xx3)+1/3#

#color(blue)(y=(58+69)/69 = 1 58/69)#
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