Problem 125 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change \((s)\) to produce a true statement. The equations \(\frac{x}{x-4}=\frac{4}{x-4}\) and \(x=4\) are equivalent.

Step-by-Step Solution

Answer

The statement is true. Both equations are equivalent.

Step 1: Analyze the first equation

In the first equation \(\frac{x}{x-4}=\frac{4}{x-4}\). As the denominators are the same, we can remove them and the equation simplifies to \(x = 4\).

Step 2: Analyze the second equation

The second equation is already simplified and it is \(x = 4\).

Step 3: Equating the simplified equations

We have \(x = 4\) from the first equation and also \(x = 4\) from the second equation, so both equations are indeed equivalent.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). In algebra, we often work with expressions to solve for the value of a variable. For example, in the exercise \(\frac{x}{x-4}\) is an algebraic expression, where x is the variable.

Understanding how to manipulate these expressions is crucial in algebra. By doing operations such as adding like terms, factoring, and expanding, we can simplify complex expressions into more manageable pieces. In the provided exercise, simplification involves recognizing that both sides of the equation have the same denominator, which allows for the cancellation of these terms and results in a straightforward equation, x = 4.

Simplifying Equations

Simplifying equations is a fundamental part of algebra that helps transform complex or long expressions into simpler forms. This process often involves combining like terms, reducing fractions, or canceling out factors. A key goal of simplification is to isolate the variable, making it easier to understand the relationship between elements in an equation.

Consider the example \(\frac{x}{x-4}=\frac{4}{x-4}\), once we identify that the denominators are identical, we can remove them, simplifying the equation to \(x = 4\). The act of simplification makes it more apparent that these two equations are indeed saying the same thing, which is they are 'equivalent equations'.

Tip: Always look for common factors or terms when you're trying to simplify an equation. As seen in the example, this can often lead to significant simplifications.

Rational Expressions

Rational expressions are fractions that involve algebraic expressions in the numerator, the denominator, or both. They are similar to rational numbers (fractions) but instead of integers, we use polynomials. It's important to mention that a rational expression is undefined when its denominator equals zero since division by zero is not possible in mathematics.

For the exercise given, \(\frac{x}{x-4}\) and \(\frac{4}{x-4}\) are both rational expressions. A key aspect of working with rational expressions is identifying restrictions for the variables. In this case, x cannot be 4 because it would make the denominator zero, leading to an undefined expression.

Identifying Equivalent Rational Expressions

Two rational expressions are considered equivalent if they simplify to the same expression. However, the given rational expressions can only be deemed equivalent if the variable does not lead to a zero in the denominator in its simplified form. The provided exercise perfectly illustrates equivalent rational expressions after simplification, leading to the same solution, x = 4.