Problem 89 - Algebra and Trigonometry

Question

\(89-96\) m State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$ \frac{16+a}{16}=1+\frac{a}{16} $$

Step-by-Step Solution

Answer

The equation is true for all values of \( a \).

Step 1: Clear the Equation Setup

Let's start by understanding the given equation: \( \frac{16+a}{16}=1+\frac{a}{16} \). We need to check if this equation holds true for all values of \( a \) by simplifying both sides.

Step 2: Simplify the Right-hand Side

On the right, we have \( 1+\frac{a}{16} \). Express 1 with a denominator of 16: \( \frac{16}{16} \). So, it becomes \( \frac{16}{16} + \frac{a}{16} = \frac{16+a}{16} \).

Step 3: Compare Both Sides of the Equation

Now both sides of the equation are \( \frac{16+a}{16} \). This shows that both sides are equivalent.

Step 4: State the Implication

Since we derived that both sides are always the same for any value of \( a \), apart from values that might make the equation undefined (denominator zero), the given equation is true for all \( a \). However, since no denominators here become zero for any finite \( a \), we safely conclude that the equation holds universally.

Key Concepts

Equation Simplification

Simplifying equations is a fundamental skill in algebra. It involves manipulating an equation to make it easier to understand or solve. In the equation given, \( \frac{16+a}{16} = 1 + \frac{a}{16} \), simplification can help determine if both sides are equal for any value of \( a \).

The process of simplification often includes combining like terms, factoring, or converting expressions to have a common denominator. In our case, one critical simplification was turning the number 1 into a fraction like \( \frac{16}{16} \), which makes it easier to compare the two fractions involved.

Simplifying not only makes it clear whether two expressions are equivalent but also helps in identifying values that make the equation true or false. In this case, once simplified to \( \frac{16+a}{16} \), it becomes apparent that the expressions are equivalent by direct comparison.

Variables

Variables are symbols used to represent numbers in algebraic expressions and equations. In the provided equation, \( a \) is the variable.

Variables are essential because they allow generalization of mathematical problems and solutions. Instead of dealing with specific numbers, variables provide a way to express relationships that hold true for a range of values. This makes mathematical models flexible and widely applicable.

When analyzing equations with variables, it's crucial to determine for which values the equation is true. If an equation simplifies to an identity (a tautology), it indicates that it's true for all permissible values of the variable, barring cases where the expression is undefined. This is particularly highlighted in our exercise as the simplification shows equality for all \( a \), subject to the conditions set by the denominators.

Denominator

The denominator is a key part of a fraction found in the lower part, below the fraction line. It indicates into how many equal parts the whole is divided. In our equation, both sides have the same denominator, 16.

Working with denominators is crucial, especially in equation simplification. Consistent denominators can make comparison straightforward since operations like addition or subtraction become easy. For example, combining \( \frac{16}{16} + \frac{a}{16} \) illustrates this by showing how both terms share a common denominator, which gets us back to \( \frac{16+a}{16} \).

It's also imperative to check when denominators make an equation invalid. Typically, an equation is undefined for values that result in a denominator of zero. However, with \( \frac{16+a}{16} \), this isn't a concern as no real number substituted for \( a \) would make the denominator zero, maintaining the equation's validity across all real numbers.