Problem 125 - College Algebra

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. The two equations \(\frac{x}{x-4}=\frac{4}{x-4}\) and \(x=4\) are equivalent.

Step 1: Solving the first equation

Begin by simplifying the first equation \(\frac{x}{x-4}=\frac{4}{x-4}\). Since the denominators are the same, then the numerators should be equal. Therefore, the equation simplifies to \(x=4\).

Step 2: Comparing the two equations

Next, compare the simplified form of the first equation, \(x=4\), with the second equation, which is also \(x=4\). They are indeed the same, therefore equivalent.

Key Concepts

Equivalent Equations

In algebra, equivalent equations are equations that, while they may look different, have the same solutions or express the same condition. To determine if two equations are equivalent, compare their solutions. If every solution to one equation is also a solution to the other, they are equivalent. For the equations given, \( \frac{x}{x-4} = \frac{4}{x-4} \) and \( x = 4 \), both can be ultimately reduced to \( x = 4 \) when solved, showing that they share the same solution. Remember, it is crucial to consider any restrictions imposed by the denominators. Thus, we conclude these equations are equivalent, excluding the point where the denominator is zero, such as when \( x = 4 \), which makes the expressions undefined.

Simplifying Equations

Simplifying equations often involves reducing them to a form that is easier to understand or solve. This usually means removing fractions or parentheses and combining like terms. When simplifying \( \frac{x}{x-4} = \frac{4}{x-4} \), notice the common denominator \( x-4 \). Since it is the same on both sides of the equation, you can set the numerators equal to each other: \( x = 4 \). Simplification helps in checking for equivalent equations and makes solving them straightforward. It’s important to ensure that the manipulations done to simplify do not violate the mathematical laws or ignore potential restrictions, such as dividing by zero. Always look for opportunities to simplify equations as a first step in problem-solving.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying rational expressions involves factoring both the numerator and the denominator and then reducing the fraction by canceling out common factors. For instance, given the expression \( \frac{x}{x-4} \), it’s clear that both parts are linear polynomials. When dealing with rational expressions, watch out for values that make the denominator zero, as these are restrictions. In our exercise, at \( x = 4 \), the expression becomes undefined because it leads to division by zero. Managing these restrictions is important because they impact the domain of the expression and the equivalency of equations involving rational expressions.