# Solving an equation using multiplication and division

Let's study a few more principles.

### Principle 4: Any number divided by itself is equal to 1.

For example:

 $\frac{2}{2} = 1$ $\frac{12}{12} = 1$ $\frac{-4}{-4} = 1$ $\frac{-8}{-8} = 1$ $\frac{3}{3x} = 1x$ $\frac{5}{5a} = 1a$ $\frac{-20}{-20y} = 1y$ $\left(\frac{-7}{-7}\right) x = 1x$

### Principle 5: One multiplied by a number is equal to the number.

For example:

 $1*8 = 8$ $1*(-5) = -5$ $1*x = x$ $m*1= m$

## Solving an equation

When solving an equation for a variable, we undo what was done to the variable. So, if 3 were multiplied by the variable, we divide both sides by 3. If -6 were multiplied by the variable, we divide by -6. We are trying to get to 1*(the variable) which equals the variable.

Remember to locate the variable and work from there. For example, in the equation 3x = 9, 3 is multiplied by the variable, so I would divide both sides of the equation by 3. In the equation -6x = 18, -6 is multiplied by the variable so I would divide both sides of the equation by -6. And, in the equation 42 = -4x, -4 is multiplied by the variable, so I would divide both sides by -4. Always divide by the exact same number to get 1.

For example:

\begin{align} 2x &= 6 \qquad \qquad \qquad & -4x &= 16 \qquad \qquad \qquad & 25 &= 6r \qquad \qquad \qquad & 30 &= -5a \\ \left(\frac{2}{2}\right) x &= \frac{6}{2} & \left(\frac{-4}{-4}\right) x &= \frac{16}{-4} & \frac{25}{6} &= \left(\frac{6}{6}\right) r & \frac{30}{-5} &= \left(\frac{-5}{-5}\right) a \\ 1*x &= 3 & 1*x &= -4 & \frac{25}{6} &= 1*r & -6 &= 1*a \\ x &= 3 & x &= -4 & \frac{25}{6} &= r & -6 &= a \\ \end{align}