91Ó°ÊÓ

Linear equations are the cornerstone of algebra and appear in forms similar to \( ax + b = c \) where \( a \) \( b \) and \( c \) are known values and \( x \) is the variable we need to solve for. The process of solving linear equations involves manipulating the equation in a way that isolates the variable on one side of the equation, thus finding its value. When faced with the equation \( -11 c = -132 \), we apply the multiplication principle, a fundamental technique in algebra that allows us to maintain the balance of an equation by performing the same operation on both sides.

The goal is to ultimately transform the equation into a simple form of \( c = \text{value} \) by eliminating coefficients and constants attached to the variable. Carefully following these steps leads to a better understanding and, thereby, correctly solving the equation. Remember, each step must maintain equality for the solution to be valid.

To isolate the variable means to get the variable alone on one side of the equation. This is often achieved by performing operations that counterbalance whatever operation is affecting the variable. In the given exercise \( -11c = -132 \), the variable \( c \) is being multiplied by -11. Our task is to undo that multiplication.

We do this by dividing both sides of the equation by -11. This operation is considered the 'opposite' or 'inverse' of multiplication, effectively cancelling out the -11 attached to \( c \). After the division, \( c \) stands alone as \( c = 12 \). This step is crucial because it simplifies the equation to its most basic form, enabling us to easily understand the value of the isolated variable.

After isolating the variable and finding a solution, it's vital to check that the solution is indeed correct. This can be accomplished by substituting the solved value back into the original equation, a method often referred to as 'equation checking' or 'solution verification'.

For our example, we insert \( c = 12 \) back into the equation \( -11c = -132 \) to get \( -11(12) = -132 \). By performing this check, we verify that both sides of the equation equal -132. This validation step confirms that the solution is accurate, and it also serves as a means to catch any potential errors made during the solving process. It's a good habit to develop as it ensures reliability and confidence in one's solutions.