91Ó°ÊÓ

Factoring is a key algebraic technique used to simplify expressions or solve equations. It involves rewriting an expression as a product of its factors. In our exercise, factoring helps identify common terms and make the equation easier to handle.
Here are some key points about factoring:

• It simplifies complex expressions.
• Identifies common factors in terms.
• Makes solving equations more straightforward.

In the given equation, a b + a = 1, we see that both terms on the left-hand side share a common factor: a. By factoring out a, we rewrite the equation as a(b + 1) = 1. This step turns the original equation into a simpler product form, allowing us to isolate and solve for a.

Isolating the variable means rearranging an equation so that the variable of interest is on one side by itself. This process simplifies the equation and reveals the solution directly.
In the given problem, we need to solve for a. Start with the factored equation a(b + 1) = 1. Our goal is to get a by itself on one side:

• We divide both sides of the equation by b + 1.
• This ensures a is isolated, giving the result:

\[ a = \frac{1}{b + 1} \]

This result shows the value of a in terms of b. By isolating a, we have effectively solved the equation. It’s a fundamental step in algebra to transition from a general equation to a specific solution.

Equivalent equations are equations that have the same solutions. When solving for a variable, we often transform the original equation into different forms that are easier to solve but still equivalent to the original equation.
In our example, we start with the equation a b + a = 1 and manipulate it to get an equivalent equation in a simpler form:

• Initially, the given equation is a b + a = 1.
• By factoring out a, we get a(b + 1) = 1, an equivalent but simpler form.

Both equations are equivalent because solving either will yield the same solution for a. This principle allows us to transform and simplify equations while ensuring we still find the correct solutions.
Finally, by isolating a, we derive the simplest form:
\[ a = \frac{1}{b + 1} \]
This transformed equation is also equivalent to the original and gives us the required solution for a.