91Ó°ÊÓ

The multiplication principle, also known as the division principle, is a fundamental concept in algebra used for solving linear equations. When we come across an equation like
\(-14a = -154\),
this principle allows us to perform the same operation on both sides of the equation without changing the solution set. In this case, we divide both sides by -14 to isolate the variable 'a'.
Here's how we apply it:
\[\frac{-14a}{-14} = \frac{-154}{-14}\]
This operation simplifies to
\(a = 11\),
because the division cancels out the -14 in front of the 'a', leaving 'a' by itself on one side of the equation. Always remember: whatever you do to one side of the equation, you must do to the other to maintain balance.

Solving linear equations follows a sequence of steps that, if executed correctly, lead to the solution. For the equation
\(-14a = -154\),
the steps are clear and systematic. First, we aim to isolate the variable, and to do that, we need to eliminate the coefficient of the variable. The most straightforward way is shown in step 1 of our example: dividing both sides of the equation by -14.
Following this initial step, we proceed to simplifying expressions, which we can see in step 2. Once simplified, if we have isolated the variable correctly, we should have a clear solution.
Finally, verification is important. Step 3 involves substituting the solution back into the original equation to confirm its validity. Remember, the order of these steps matters to solve the equation efficiently and accurately.

Simplifying expressions is a process that makes equations easier to understand and solve. It involves reducing equations to their simplest form. After dividing both sides of our equation by -14, we are left with a simplified expression:
\(a = 11\).
Simplification in this context is straightforward, but sometimes we have to perform additional operations like combining like terms or distributing multiplications over additions.
It's important to work methodically to avoid mistakes. Simplifying expressions not only helps to solve equations but also prepares the foundation for higher-level algebra, where expressions can become quite complex.

Checking the solution is an essential step in solving equations, often overlooked in the rush to complete a problem. For our example, after arriving at the solution \(a = 11\), we substitute this value back into the original equation to verify its correctness:
\[-14(11) = -154\].
Carrying out the multiplication, we confirm that both sides of the equation are indeed equal:
\[-154 = -154\].
If the two sides match, as they do here, we can confidently say that the solution is correct. Solution checking not only helps to confirm the answer but also reinforces understanding of the underlying concepts.