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When solving equations, the main goal is to find the value of the unknown variable that makes the equation true. Equations are mathematical statements that show the equality between two expressions. The process typically involves manipulating the equation to "isolate" the variable on one side. This way, you can clearly see the solution.

For example, consider the equation given in the exercise: \(b - 5 = -19\). Solving this involves adding, subtracting, multiplying, or dividing both sides of the equation with the same number. This is done to eliminate terms joined with the variable, exposing the variable by itself.

• Start by identifying the operation that binds the variable. In this case, 'b' is subtracted by 5.
• Apply the inverse operation to both sides to balance the equation. Here, you add 5 to remove the subtraction.

By keeping equations balanced, you ensure that both sides remain equal as adjustments are made.

Isolating the variable is a crucial step in solving equations. It means rearranging the equation so the variable is by itself on one side. This simplification makes it easy to read the solution.

For the equation \(b - 5 = -19\), isolating the variable 'b' requires you to eliminate the -5. You achieve this by doing the opposite of subtraction, which is addition. So you add 5 to both sides:
\[b - 5 + 5 = -19 + 5\]

• On the left, the -5 and +5 cancel each other, which leaves 'b' isolated.
• On the right side, perform the arithmetic to simplify: \(-19 + 5 = -14\).

Now 'b' is isolated, revealing that its value is -14. This straightforward method is effective for unlocking the mystery behind the unknown in a simple equation.

Verifying your solution is an essential part of solving equations. It confirms the accuracy of your answer and builds confidence in the method used. After isolating the variable and determining its value, substitute it back into the original equation to ensure it holds true.

For instance, with the solution \(b = -14\), substitute back into the original equation:
\(-14 - 5 = -19\)

• If both sides of the equation match, then the solution is correct.
• In our example, \(-19 = -19\), which confirms that \(b = -14\) is indeed the correct solution.

This step not only verifies your answer but also shows you understand the operations involved in the equation.