Question

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

Answer 1

If you start to use decimals you will introduce errors. keep to fractions!
`color(blue)(x=58/23" "y=1 58/69)`

Explanation:

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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