Question

What is the solution of the system of equations: `-6x+2y=-8` ?

Answer 1

`(x, y) in { (t, 3t-4) : t in RR }`

Explanation:

This system of equations only has one linear equation in two unknowns and hence an infinite number of solutions.

The solutions lie along the line described by the given equation:

`-6x+2y=-8`

We can rearrange this equation to express `x` in terms of `y` or `y` in terms of `x` as follows:

Divide both sides of the equation by `2` to get:

`-3x+y = -4`

Add `3x` to both sides to get:

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

For any value of `x` this gives us the corresponding value of `y`.

This formula is in the form:

`y = mx+c`

known as slope intercept format, where `m=3` is the slope of the line and `c=-4` is the `y` intercept.

If we add `4` to both sides and transpose we get:

`3x = y+4`

Then dividing both sides by `3` we get:

`color(blue)(x = 1/3y+4/3)`

For any given `y`, this formula gives us the corresponding value of `x`.

Alternatively, we can use the previous slope intercept format equation to derive a parametric representation of the line as:

`(t, 3t-4)`

where `t in RR`

So we can express the solution space of the original system of equation(s) as:

`color(blue)((x, y) in { (t, 3t-4) : t in RR })`

graph{y=3x-4 [-9.42, 10.58, -5.72, 4.28]}