Question

How do you solve `x/3=-10`?

Answer 1

`x = -30`

Explanation:

When dealing with equalities, you can multiply both sides by a same number, and the inequality will preserve its truth value. By this, I mean that if you start with a true equality, it will still be true, for example

`3 = 3 \to 3*4 = 3*4 \to 12=12`

So, we started with a true equality (`3=3`), and we multiplied both sides by `4`, obtaining another true identity (`12=12`).

On the other hand, if the two sides are different, they will still differ after being multiplied by the same number:

`3 \ne 2 \to 3*6 \ne 2*6 \to 18 \ne 12`

So, we started with a false equality (`3\ne 2`), and we multiplied both sides by `6`, obtaining another true identity (`18 \ne 12`).

In your case, you only need to multiply both sides by `3`: the expression becomes

`3*\frac{x}{3} = -10*3`

Why did we choose `3`? Because our goals is to isolate the `x` on the left side, obtaining an expression like `x=...`. And the `3` we multiplied by simplifies with the `3` at the denominator, allowing us to reach our goal:

`cancel(3)*\frac{x}{cancel(3)} = -10*3`

Of, course, this comes with the price of an added calculation on the right hand side, but that's not much of a problem:

`x = -10*3 = -30`

Equation solved! In fact, we reached a form like `x=k`, for some real number `k`, which means that that particular value is the solution for the equation.

Answer 2

`x=-30`

Explanation:

Given: `x/3=-10`

The objective is to end up with just one `x` and for it to be on its own on one side of the = and everything else on the other side.

To end up with just `x` on the left we change the `1/3` into 1. This is because `1xx x` is just `x`

`color(green)(x/3=-10 color(white)("dddd") ->color(white)("dddd") x xx1/3=-10`

Multiply both sides by `color(red)(3)`

`color(green)( color(white)("dddddddddddd") ->color(white)("dddd") x xx1/3color(red)(xx3)=-10color(red)(xx3)`

`color(green)( color(white)("dddddddddddd") ->color(white)("dddd") x color(white)("dd")xx3/3color(white)(".d")=-30)`

But `3/3` is the same value as 1 giving: `color(white)("dd")x=-30`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("General shortcut rule")`

To move a value to the other side of the equals:

For addition or subtraction:
`color(white)("dd")`Move to the other side and apply the opposite action. Addition is `color(white)("dd")`changed to subtraction and subtraction is changed to addition.

For division or subtraction:
`color(white)("dd")`Move to the other side and apply the opposite action. Division is `color(white)("dd")`changed to multiplication and multiplication is changed to
`color(white)(".d")` division.