Problem 35 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Simplify Both Sides

The equation given is \( a^2 x + (a - 1) = (a + 1)x \). Let's simplify both sides of the equation by expanding and rearranging terms where necessary.

Step 2: Move All Terms Involving x to the Left

Subtract \((a+1)x\) from both sides to collect all x terms on one side: \(a^2 x - (a+1)x + (a - 1) = 0\).

Step 3: Factor x out of the Left Side

Factor out \(x\) from the left-hand side: \(x(a^2 - (a+1)) = 1 - a\). This simplifies to \(x(a^2 - a - 1) = a - 1\).

Step 4: Solve for x

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

Key Concepts

Equation Simplification

Simplifying equations is the first key step in solving them efficiently. It involves reducing the equation to its simplest form, which makes it easier to observe what needs to be done next. In this equation: \[ a^2 x + (a - 1) = (a + 1) x \]our goal is to rewrite it in a simpler form so we can easily solve for the variable. **Steps to Simplify an Equation:**

• Remove Parentheses: Expand any terms in the equation, though in this case, there are no parentheses to expand.
• Rearrangement: Swap terms on either side of the equation if needed, trying to position different components logically.
• Combine Similar Operations: Ensure that each component of the equation is optimized for the operations involved.

By simplifying, you reduce potential errors that come from working with a more complex equation.

Collecting Like Terms

After simplifying the equation, the next important task is to collect like terms. Like terms have the same variable parts; in this context, all terms with the variable \(x\) need to be grouped together for seamless solving. **Example: Grouping Terms with \(x\)**For the given equation:After the simplification steps, you notice:\[ a^2 x - (a + 1)x + (a - 1) = 0 \]Identify terms with \(x\), subtract them together to rearrange:\[ (a^2 - (a + 1))x = 1 - a \]**Benefits of Collecting Like Terms:**

• Simplifies the equation structure.
• Reduces the possibility of calculation errors.
• Makes the equation appearance cleaner and aligns terms properly.

Factoring Linear Expressions

Factoring is a crucial operation in simplifying expressions and equations. When solving linear equations, factoring allows the expression to be broken into parts that are easier to handle. **Steps to Factor:**Take the rearranged part of the equation involving \(x\):\[ x(a^2 - a - 1) = a - 1 \]Factor the left side by recognizing that \(x\) is a common factor:\[ x(a^2 - a - 1) \]This form indicates that \(x\), multiplied by the expression \(a^2 - a - 1\), equals the term on the other side of the equation. **Why Factor?**

• Factoring makes solving for \(x\) straightforward—either adjust the value on the equation or recognize a solution quickly without further expansion.
• Allows easy substitution and verification with solutions.
• Aids in simplifying the algebraic manipulation required in future steps.