91Ó°ÊÓ

In algebra, the process of making a variable stand alone on one side of an equation is known as 'isolating the variable'. This is often the essential goal when we want to solve for a specific unknown, and it's what allows us to find the value of a variable in relation to others.

For example, in the exercise provided, we want to solve for the variable 'c'. Isolating 'c' involves performing operations that simplify the equation so that 'c' is by itself on one side. This often requires moving other terms to the opposite side of the equals sign. In the exercise, we do this by first distributing (which we will talk about next), and then by moving all terms not containing 'c' away from 'c'. This is done by subtracting 'a' from both sides of the equation, making progress towards isolating 'c'.

Remember to always do the same operation on both sides of the equation to maintain balance, which is a fundamental rule in algebra.

The distribution property, also known as the distributive law of multiplication, is a cornerstone of algebra that allows us to multiply a single term by each term within a parenthesis. The formula for this property is \( a(b+c) = ab + ac \).

In our exercise, we apply this property in the beginning to remove the parentheses. The initial equation \( L = a(1+ct) \) becomes \( L = a + act \) after applying the distribution property. This step is crucial -- it simplifies the equation and prepares it for further manipulation to isolate the variable, 'c'.

It is important to distribute correctly to ensure all terms within the parentheses are multiplied by the factor outside, which in our case is 'a'. Failure to distribute properly is a common mistake that can lead to incorrect solutions.

Algebraic manipulation involves rearranging and simplifying algebraic equations using a set of rules and properties, like the distributive property, to find the value of an unknown variable. Mastery of algebraic manipulation enables you to solve a wide variety of equations.

In the step-by-step solution provided, once we have distributed 'a' and subtracted it from both sides, we are left with \( L - a = act \). To isolate 'c', we need to perform further algebraic manipulation. The next step is to divide both sides of the equation by the product 'at': \( \frac{L - a}{at} = \frac{act}{at} \). This cancels 'at' on the right side, leaving us with the isolated variable: \( c = \frac{L - a}{at} \).

This demonstrates a sequential application of algebraic operations – distributing, subtracting, and dividing, each an example of algebraic manipulation, to simplify the equation and resolve it for the desired variable.