Question

How do you solve this system of equations: `x- 4y \geq 4; 4y - x > 8`?

Answer 1

The equations are inconsistent and there is no solution.

Explanation:

The lines are parallel so there is no value `(x,y)` that will satisfy both equations.

They will never have a point in common because they will never intersect.

Solve both equations for `y` to put them in slope-intercept form

`x - 4y >=4`   Solve for `y`

1) Subtract `x` from both sides to isolate the `-4y` term
`- 4y >= -x + 4`

2) Divide both sides by `-4` to isolate `y`
`y >=(x)/(4) - 1`

`"Slope is"   (1)/(4)`

`color(white)(mmmmm)`―――――――

`4y - x > 8`   Solve for `y`

1) Add `x` to both sides to isolate the `4y` term
`4y > x + 8`

2) Divide all the terms on both sides by `4` to isolate `y`
`y > (x)/(4) - 2`

`"Slope is also"   (1)/(4)`

`color(white)(mmmmm)`―――――――

Because the slopes are the same, the lines are parallel.

That means that they will never have an intersection point, a point that satisfies both equations.

`color(white)(mmmmm)`―――――――

Check

Solve the equations as simultaneous equations to see if that gives a value for `y`

`x - 4y >=4`
`4y - x > 8`

- same as -

`- 4y + x >=  4`
`color(white)(m)` `4y - x >   8`
`color(white)()`――――――――
`color(white)(.......)` `0    >= 12`

This is not true, so the equations are inconsistent.

`Check`