Problem 85 - Algebra and Trigonometry

Question

\(79-92\) Solve the equation for the indicated variable. $$ a^{2} x+(a-1)=(a+1) x ; \text { for } x $$

Step-by-Step Solution

Answer

\( x = \frac{-(a - 1)}{a^2 - a - 1} \)

Step 1: Identify the Equation

We begin with the equation: \( a^2 x + (a - 1) = (a + 1) x \). Our task is to solve for the variable \( x \).

Step 2: Rearrange the Equation

Move all terms involving \( x \) to one side of the equation: \( a^2 x - (a + 1) x = -(a - 1) \).

Step 3: Factor Out \( x \)

Factor \( x \) from the left side of the equation: \( x(a^2 - (a + 1)) = -(a - 1) \).

Step 4: Simplify the Equation

Simplify the expression \( a^2 - (a + 1) \) to get \( x(a^2 - a - 1) = -(a - 1) \).

Step 5: Solve for \( x \)

Divide both sides by \( a^2 - a - 1 \) to isolate \( x \): \( x = \frac{-(a - 1)}{a^2 - a - 1} \).

Key Concepts

Solving Equations

Solving equations is a process where we aim to find the value of a variable that makes the equation true. In algebra, equations are like balance scales. Each side of the equation must balance, which means they must be equal. To achieve this balance when solving for a specific variable, we perform operations to both sides of the equation to keep it true.
We often start with identifying the equation and moving terms around to isolate the variable we're interested in. Ensuring that every step maintains the balance is crucial. In this process, we rely on basic algebraic operations: addition, subtraction, multiplication, and division.

• Add or subtract terms to move them from one side of the equation to the other.
• Multiply or divide terms to further simplify or maintain balance.

Our main goal is to manipulate the equation so that the variable we want to solve for is on one side by itself.

Variable Isolation

Variable isolation means getting the variable alone on one side of the equation. This is often achieved through a series of algebraic manipulations, like rearranging terms, so the variable is by itself.
In our original problem, the main goal is to rearrange the equation to form something like "variable = expression." To do this, we need to move all terms involving the variable to one side.
This can involve:

• Addition or subtraction to eliminate constant terms from the side with the variable.
• Combining like terms to simplify.
• Using multiplication or division to simplify down to just one instance of the variable.

Once the variable is isolated, we can see clearly what it equals, allowing us to "solve" or determine its value.

Factoring

Factoring is an algebraic process used to simplify expressions. It involves expressing a mathematical expression as a product of its factors. In the context of solving algebraic equations, factoring facilitates isolating the variable.
In the given problem, once you gather all terms involving the variable on one side, you may notice that they share a common factor. Factoring this common variable out means rewriting the expression in terms of a product, which helps simplify it. For instance, the terms on one side might both involve \( x \).
Factoring is an essential step that allows for simpler arithmetic and clearer isolation of the variable, which is precisely what makes the problem easier to solve.

Simplifying Expressions

Simplifying expressions involves making them easier to work with. In algebra, simpler expressions help us more easily find a solution. Simplifying can involve combining like terms, reducing fractions, or generally rewriting an expression in a simpler form.
For example, in our problem, the expression \( a^2 - (a + 1) \) was simplified to \( a^2 - a - 1 \). This step is crucial because it makes the subsequent steps more manageable and prevents errors during variable isolation and solving.
When simplifying, look out for:

• Distributive properties to unleash hidden expressions.
• Opportunities to combine terms through addition or subtraction.
• Factoring any complex terms to reduce them.

Ultimately, simplifying sets the stage for efficiently solving the equation, ensuring that your final answer is neat and clear.