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When solving equations, one of the key steps is isolating the variable you are solving for. Let's consider the example equation: 84 = -12zYour goal is to get 'z' by itself on one side of the equation. To do this, you must 'undo' any operation that is being performed on 'z'. This usually involves using the inverse operation. For instance, if 'z' is being multiplied by -12, you'll need to perform the opposite operation, which is division, to isolate 'z'. Once isolated, you can easily understand the value of the variable.

In order to isolate a variable, you often need to divide both sides of an equation by a particular number. This ensures that the equation stays balanced. In our example:84 = -12zWe divide both sides by -12 to isolate 'z'. This can be written as:\[84 \text{\div} -12 = -12z \text{\div} -12\]The division on both sides simplifies the equation and isolates 'z'. After performing the division:\[84 \text{\div} -12 = -7\]Thus, we get:\[-7 = z\]or:\[z = -7\]

Once you have divided both sides of the equation, the next step is simplifying. Simplifying involves performing the division or any other arithmetic necessary and writing the simplest form of the equation. In our case:\[84 \text{\div} -12 = -7\]After simplifying, we get a clear and straightforward result that tells us the value of 'z' is -7. This final step ensures that the variable is in its simplest possible form, making it easy to interpret the solution. In summary:\[-7 = z\] or equivalently:\[z = -7\] You have successfully solved the equation by isolating the variable, dividing both sides, and simplifying the equation. By mastering these core concepts, you can confidently tackle any linear equation.