Question

How do you solve `-4+ 6x = - 3x - 31`?

Answer 1

`x=-3`

Full explanation given. The maths on its own is very fast when you get used to the shortcuts. Much less writing.

Explanation:

`color(blue)("VERY IMPORTANT FACT")`

Sorry if I am stating the obvious but bear with me.

The breakthrough for this type of thing is the realisation that all 'the bits' in an 'equation' have value.

The first very important thing is to understand the = sign.

Note the similarity 'equation -> equals'

If something is equal to something else they are the same.

If when you change something on one side of the equals and it changes the 'value' you must change the other side in the same way if you wish to maintain equality.

This can be used to advantage

By convention

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`color(blue)("Answering the question")`

By example; it is convention that a count of 6 letters of `x` is `6xx x" "` which is written as `6x`.

It is understood but not written that there is a multiply between the 6 and the `x`

Let `x` be some number the value of which we do not yet know.

Given:`" "-4+6x=-3x-31`

Add `color(red)(4)` to both sides

`color(green)(-4+6x=-3x-31" "->" "-4color(red)(+4)+6x=-3xcolor(red)(+4)-31`

`color(green)(" "->" "0color(white)(..)+6x=-3x-27`

Notice that this has actually moved the 4 from the left to the right. In doing so it changed the sign from minus to plus.

Add `color(red)(3x)` to both sides

`color(green)(6x=-3x-27" "->" "6xcolor(red)(+3x)=-3xcolor(red)(+3x)-27 )`

`color(green)(" "->" "9xcolor(white)(..)=" "0" "-27)`

We have moved the `3x` to the other side and changed its sign

Divide both sides by `color(red)(9)`

`color(green)(9x=-27" "->" "9/(color(red)(9))xx x=(-27)/(color(red)(9))`

`color(green)(" "->" "1 xx x=-27/9)`

We have moved the 9 to the other side and changed it from multiply to divide which is its opposite action.

`x=-(27-:9)/(9-:9)`

`x=-3/1 = -9`