Question

How do you solve this system of equations: `3y + 6x = - 24 and z = - 2y - 16`?

Answer 1

See below.

Explanation:

Notice we have three variables and two equations. In order for a `color(blue) "Linear System"` to have a unique solution, we must have the following:

For n number of variables we need n number of consistent, linearly independent equations.

We will not concern ourselves with consistency and independence now

If we have n number of variables and n-1 equations the we will have to solve in terms of one variable. For n-2 equations, in terms of two variables etc.

From example we have `n=3` variables and `3-1` equations, so we will be solving in terms of one variable.

`color(blue)(3y+6x=-24color(white)(888)[1]`

`color(blue)(z=-2y-16color(white)(8888)[2]`

Lets find `z` in terms of `x`.

Using `[1]`

`3y+6x=-24`

`3y=-24-6x`

`color(blue)(y=-8-2xcolor(white)(888)[3]`

Substituting `[3]` in `[2]`

`z=-2(-8-2x)-16`

`z=16+4x-16`

`z=4x`

Now find `y` in terms of `x`.

This is equation `[3]` we found earlier.

So we have:

`y=-8x-2x`

`z=4x`

We can write our solutions in the following form:

`( x ,y ,z)`

Then for arbitrary `x` ( we assign any real value to `x` )

`color(blue)((x,-8-2x,4x)`

For `x` being a real number we have an infinite number of solutions.