Question

What is the solution to the equation `3x-6=-2x+9`?

Answer 1

`x = 3`

Explanation:

It is necessary to manipulate the equation so that only `x` is on the left hand side.

Progress is made by choosing ways to work towards this, ensuring that anything that is done to one side of the equation is also done to the other.

To make a start, it is possible to add `2x` to the right hand side of the equation, because the term in `x` that is there is `-2x`, and `2x - 2x = 0`, which gets rid of the `-2x` from that side. It is necessary to do the same thing to the other side of the equation.

That is

`3x - 6 = -2x + 9`

implies

`3x + 2x - 6 = -2x + 2x + 9`

Collecting terms on both sides

`5x - 6 = 9`

Similarly, it is possible to get rid of the `-6` on the left hand side (where I am trying to have only terms in `x`) by adding `6`. As before, if I do it to one side of the equation, I have to do the same thing to the other side.

So,

`5x - 6 = 9`

implies

`5x - 6 + 9 = 9 + 6`

That is

`5x = 15`

To make further progress, it is necessary to change `5x` into `x` and this can be done by dividing by `5`. I have to do the same thing to the other side.

So,

`5x = 15`

implies

`(5x)/5 = 15/5`

That is

`x = 3`

and this is the solution!

Revisiting the steps, you might note that adding the same thing to both sides, with the thing to be added being chosen so that it makes one of the terms on one side of the equation disappear by summing to zero, is equivalent to the rule "if something is moved from one side of the equation to the other, it is necessary to change its sign".

So I could have "just" moved the `-2x` from the right hand side of the equation to the left hand side, changing it to `2x` as I do so.

That is

`3x - 6 = -2x + 9`

Implies

`3x + 2x - 6 = 9`

which implies

`5x - 6 = 9`

I could have done the same thing for the `-6` on the left hand side of the equation, that is

`= 5x - 6 = 9`

implies

`= 5x = 9 + 6`

which implies

`5x = 15`

then proceed as before (with division of both sides by `5`)

Finally, always check the answer if it is possible to do so!

When `x = 3`,

the left hand side of the equation is
`3 xx 3 - 6 = 9 - 6 = 3`

and the right hand side of the equation is
`-2 xx 3 + 9 = - 6 + 9 = 3`

Both sides of the equation are the same so the answer is correct.