Problem 58 - College Algebra

Question

In Exercises \(51-58,\) determine whether each equation is an identity, a conditional equation, or an inconsistent equation. $$ \frac{3}{x-3}=\frac{x}{x-3}+3 $$

Step-by-Step Solution

Answer

The given equation is a conditional equation. The equation is true when \(x=3\).

Step 1: Simplify the equation

First, recognize that on each side of the equation, the expressions have a common factor, \(x-3\). Therefore, with \(x \neq 3\) to avoid division by zero, both sides can be multiplied by \((x-3)\) to eliminate the denominator. Doing so gives:\n\(3 = x + 3(x - 3)\)

Step 2: Further simplify the equation

Expand out the right-hand side: hence\n\(3 = x + 3x - 9\) . \nThis simplifies to: \n\(3 = 4x - 9\)

Step 3: Identify the type of equation

The equation \(3=4x-9\) is true for certain values of \(x\) and is not always true or never true. For example, and specifically, it is true for \(x=3\), since \(3=4*3-9\). Therefore, this is a conditional equation. The condition here is that it's only true when \(x=3\).

Key Concepts

Identity Equation

An identity equation is a type of equation that is always true for any value of the variable involved. This means that no matter what number you substitute for the variable, the equation remains valid. A great way to recognize an identity equation is to look for equations where, when simplified, you end up with a statement like \(0 = 0\), or \(x = x\). These illustrate that the equation holds under all circumstances.Identity equations are special because they depict inherent truths in algebra. They often arise in simplifying expressions or solving equations where, after all the transformations, both sides of the equation are completely equal in every scenario. For example, the equation \(x + 2 = x + 2\) is an identity because it holds for any possible value of \(x\).

Conditional Equation

A conditional equation is one that is only true for certain values of the variable. When you simplify a conditional equation, you often get something like \(x = a\), where \(a\) is a specific number. This means the equation only holds when \(x\) takes this particular value.From the original exercise, we determined that the equation \(3 = 4x - 9\) is conditional. When solved, it shows that it only holds true when \(x = 3\). This specificity makes conditional equations intriguing because they require finding the precise solution or solutions that satisfy the equation. Unlike identity equations, you can think of conditional equations as having a unique fix, or a limited set of fixes.These equations commonly arise in both simple and complex algebraic problems where you are tasked with finding that exact number or numbers that make the equation true. It nurtures problem-solving skills as it demands that students learn techniques to isolate variables and solve for specific values.

Inconsistent Equation

Inconsistent equations are equations that have no solutions. This means there is no possible value for the variable that will make the equation true. In algebra, inconsistent equations often arise in the process of solving what initially appears to be a straightforward equation.When you simplify such an equation, you might end up with a statement like \(5 = 0\) or anything else that clearly conveys a falsehood. Since this outcome is illogical, it signals that the original equation cannot be true under any circumstance.An example might be \(x + 3 = x + 5\). When simplified, you subtract \(x\) from both sides and end up with \(3 = 5\), which is clearly inconsistent.Inconsistent equations highlight constraints and limitations within algebraic operations. Recognizing them helps students understand that equations must always start from a logical point, and their solutions should remain within the realm of what's possible.