Question

How do you solve the system of equations `2x+8y=6` and `-5x-20y=-15`?

Answer 1

This system of equation doesn't have any solutions.

Explanation:

`2x+8y=6`
`-5x-20y=-15`

Let's simplify these equations: divide the first one by `2` andd the second by `-5`:

`x+4y=3`
`x+4y=15`

Now, using any of the equations, we must express one variable by the other, let's say `x` by `y` from the second equation:

`x=15-4y`

Using this we solve the other equation:

`x+4y=3`
`15-4y+4y=3`
`15=3`

We got `15=3` which obviously isn't true so this equation has no solution. It follows that this system of equation doesn't have any solutions as well.

Answer 2

`x` can have any value and the equations will work.
The y- values will depend on the x chosen.

Explanation:

`2x+8y =6" and "-5x -20y = -15`

There are different options available to solve the equations.
`rarr" "` elimination
`rarr" "` substitution `rarr` single variable
`rarr" "` equating
`rarr" "`matrices
`rarr" "` graphically

To be able to use substitution, a single variable in one of the equations is a useful indicator

`2x+8y =6 " "div2 rarr " "x+4y=3`

Make x the subject of the equation:

`color(red)(x = 3-4y)`

Replace `color(red)x` in the second equation by `color(red)((3-4y))`

`" "-5color(red)(x) -20y = -15`
`color(white)(xxxx)darr`
`-5color(red)((3-4y))-20y =-15`

`-15+20y -20y = -15`

`-15 = -15`

`x` can have any value