Problem 89 - Algebra and Trigonometry

Question

Solve the equation for the indicated variable. \(a^{2} X+(a-1)=(a+1) x, \quad\) for \(x\)

Step-by-Step Solution

Answer

The solution for \(x\) is \(x = \frac{1-a}{a^2 - a - 1}\).

Step 1: Simplify the Equation

Start with the given equation: \( a^2 X + (a - 1) = (a + 1) x \). Notice that the terms \((a^2 X)\) and \((a + 1)x\) both contain the variable \(x\), and our goal is to solve for \(x\).

Step 2: Isolate the Terms Involving x

Subtract \((a + 1)x\) from both sides of the equation to get: \( a^2 X - (a + 1)x + (a - 1) = 0 \). This step is to bring all terms involving \(x\) on one side of the equation.

Step 3: Combine Like Terms

Group and factor out \(x\) from the left-hand side: \( (a^2 - (a + 1))x + (a - 1) = 0 \). Simplify this as: \( (a^2 - a - 1)x + (a - 1) = 0 \).

Step 4: Solve for x

Rearrange the equation to solve for \(x\): \[ x = \frac{1-a}{a^2 - a - 1} \]. Ensure that \(a\) is such that the denominator is not zero.

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to simplify or solve them. In this instance, it is crucial to look at the forms of the expressions on both sides of the equation. Consider the given equation:

• Start with: \( a^2 X + (a - 1) = (a + 1) x \).
• The key is to manipulate this into a clearer form where both sides are balanced and simplified.
• Notice that we can perform operations like adding, subtracting, multiplying, or dividing by the same number on both sides to maintain equality.

By approaching the equation with these operations, you can methodically simplify and move towards solving for the unknown variable, which in this case, is \(x\).

Isolating Variables

Isolating the variable means getting the variable you're solving for alone on one side of the equation. Here, you need to isolate \(x\) by gathering all terms containing it on one side and all constant terms on the other:

• Subtract \((a + 1)x\) from both sides to begin gathering \(x\).
• This results in rearranging to: \( a^2 X - (a + 1)x + (a - 1) = 0 \).
• Now it is easier to factor out the \(x\) terms which contribute directly to isolating the variable.

Through this method, you efficiently focus on rearranging the equation so that \(x\) is more accessible, simplifying the path to finding its value.

Factoring Expressions

Factoring is breaking down an expression into simpler components that can be easily worked with. In our equation, after gathering like terms:

• You recognize that \( (a^2 - a - 1)x + (a - 1) = 0 \) needs to be addressed.
• To factor, look for common terms that can be extracted or "factored out", such as the \(x\) in \( (a^2 - a - 1)x \).
• By factoring out \(x\), the equation becomes more straightforward: \( x \cdot (a^2 - a - 1) = 1 - a \).

This step simplifies solving by reducing complex expressions into simpler terms that are easier to interpret and solve.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. In the solution, when solving for \(x\), the equation becomes a rational expression:

• The expression for \(x\) is: \( x = \frac{1 - a}{a^2 - a - 1} \).
• Understanding the nature of rational expressions is crucial; both the numerator and the denominator have their respective roles.
• It's important to ensure the denominator is not zero, as this would make the expression undefined.

Analyzing the components of rational expressions helps understand their behavior and the constraints, leading to correct and valid solutions. Always check for any restrictions, such as values of \(a\) that might make the denominator zero to avoid invalid solutions.