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

Jun 14, 2018

$x = - 30$

#### 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

$3 = 3 \setminus \to 3 \cdot 4 = 3 \cdot 4 \setminus \to 12 = 12$

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:

$3 \setminus \ne 2 \setminus \to 3 \cdot 6 \setminus \ne 2 \cdot 6 \setminus \to 18 \setminus \ne 12$

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

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

$3 \cdot \setminus \frac{x}{3} = - 10 \cdot 3$

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

$\cancel{3} \cdot \setminus \frac{x}{\cancel{3}} = - 10 \cdot 3$

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

$x = - 10 \cdot 3 = - 30$

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.

Jun 14, 2018

$x = - 30$

#### Explanation:

Given: $\frac{x}{3} = - 10$

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 $\frac{1}{3}$ into 1. This is because $1 \times x$ is just $x$

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

Multiply both sides by $\textcolor{red}{3}$

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

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dddddddddddd") ->color(white)("dddd") x color(white)("dd")xx3/3color(white)(".d}} = - 30}$

But $\frac{3}{3}$ is the same value as 1 giving: $\textcolor{w h i t e}{\text{dd}} x = - 30$
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$\textcolor{b l u e}{\text{General shortcut rule}}$

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

 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.
$\textcolor{w h i t e}{\text{dd}}$Move to the other side and apply the opposite action. Division is $\textcolor{w h i t e}{\text{dd}}$changed to multiplication and multiplication is changed to
$\textcolor{w h i t e}{\text{.d}}$ division.