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In any equation, our goal is to find the value of the variable, like 'p' in this case. But first, we need to 'isolate the variable.' This means we want the variable term to be by itself on one side of the equation.
To do this, identify the term that contains the variable. Here, it is '8p' in the equation \(8p + 7 = 47\).
Next, we need to remove any constants (numbers) that are added or subtracted to this term. This often involves doing the opposite operation. For instance, if a constant is added, we subtract it from both sides.
For our example, we subtract 7 from both sides, which gives us: \(8p + 7 - 7 = 47 - 7\).
Remember, whatever you do to one side of the equation, you must do to the other side to keep it balanced.

Once you have isolated the variable term, you need to simplify both sides of the equation. Simplifying means performing all operations to condense the expression. This makes the equation easier to solve.
So if we follow from our example, after isolating the variable we get: \(8p + 7 - 7 = 47 - 7\). Simplifying this gives us: \[8p = 40\]
Notice how we combined like terms and simplified each side of the equation. This is an important step because a simplified equation is much easier to solve.
Always double-check to make sure you have combined all like terms and simplified correctly.

Now, with the equation simplified, the final step is to 'solve for the variable.' This means performing operations to isolate the variable completely and find its value.
From our current equation \[8p = 40\], we need to solve for 'p.' To do this, perform the opposite operation to what is currently being done to the variable.
Here, '8p' means '8 times p.' The opposite is division. So, we divide both sides of the equation by 8: \[p = \frac{40}{8}\]
Simplifying the right-hand side gives us: \(p = 5\)
Always check your work by plugging the value back into the original equation to ensure both sides are equal.
By following these steps to isolate the variable, simplify both sides, and solve for the variable, you can tackle any similar equation with confidence.