Problem 42 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ \frac{1}{3-t}+\frac{4}{3+t}+\frac{16}{9-t^{2}}=0 $$

Step-by-Step Solution

Answer

The solution is \(t = \frac{31}{3}\) and it is valid.

Step 1: Simplify the Equation

Recognize that the term \(9-t^2\) in the equation can be factored as \((3-t)(3+t)\). Rewrite the equation using this factorization:\[ \frac{1}{3-t} + \frac{4}{3+t} + \frac{16}{(3-t)(3+t)} = 0 \]

Step 2: Find Common Denominator

The common denominator for all fractions is \((3-t)(3+t)\). Rewrite each term using this denominator:\[\frac{1 (3+t)}{(3-t)(3+t)} + \frac{4 (3-t)}{(3-t)(3+t)} + \frac{16}{(3-t)(3+t)} = 0\]This simplifies to:\[\frac{3+t + 4(3-t) + 16}{(3-t)(3+t)} = 0 \]

Step 3: Simplify the Numerator

Combine all the terms in the numerator:\[3 + t + 12 - 4t + 16 = 0 \]Simplify to:\[31 - 3t = 0\]

Step 4: Solve for t

Solve the simplified equation \(31 - 3t = 0\) for \(t\). Rearrange the equation to isolate \(t\):\[31 = 3t \]Divide both sides by 3:\[t = \frac{31}{3}\]

Step 5: Check the Solution

Verify that \(t = \frac{31}{3}\) is a valid solution by substituting it back into the original equation and ensure no division by zero occurs.Since \(t = \frac{31}{3}\) does not make any of the denominators zero, it is a valid solution.

Key Concepts

Factoring

Factoring is an essential mathematical skill. It helps break down complex equations into simpler parts. In the equation given, identifying how to factor the expression is vital. We have the expression \(9 - t^2\). This is a difference of squares. The formula for factoring a difference of squares is \(a^2 - b^2 = (a-b)(a+b)\). By recognizing this, we can transform \(9-t^2\) into \((3-t)(3+t)\). This refactors the equation into a more workable form that simplifies further operations.

Common Denominator

When dealing with fractions in an equation, finding a common denominator is crucial. It enables you to combine the fractions effectively. In this exercise, after factoring, the common denominator for all the terms is \((3-t)(3+t)\). By rewriting each fraction with this common denominator, we align the equation. This step is important because it allows every term to be expressed as parts of the same whole. Thus, we can easily combine these terms, simplifying the problem significantly.

Imagine trying to add fractions with different denominators without a common base. It's almost impossible. By creating a common framework, like a shared denominator, fractions are easy to add or subtract.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This means combining like terms and reducing numerical fractions. Once everything is under a common denominator, the next step is to simplify.

In our exercise, the equation becomes \(\frac{3+t+4(3-t)+16}{(3-t)(3+t)} = 0\). The focus here is on the numerator. Combine all parts of the numerator: \(3 + t + 12 - 4t + 16\). This simplifies to \(31 - 3t\). Removing like terms helps concentrate the equation into its most basic and understandable form. This simplification paves the way for solving the equation efficiently.

Solving Equations

Solving equations involves finding the value of unknowns that satisfy the equation. After simplifying the fractional equation, we're left with \(31 - 3t = 0\). Solving for \(t\), we rearrange to get \(31 = 3t\). Dividing both sides by 3 gives us the solution: \(t = \frac{31}{3}\).

This method shows how important each previous step is. Without proper factoring, finding a common denominator, and simplifying fractions, reaching this solution would be challenging. Always check that the calculated solution doesn't lead to undefined operations, like division by zero, which confirms the solution's validity.