Problem 2 - Algebra and Trigonometry Real Mathematics, Real People

Question

A(n) _______ solution does not satisfy the original equation.

Step-by-Step Solution

Answer

Extraneous

Step 1: Understanding Mathematical Solutions

In mathematics, a 'solution' is a value or a set of values that satisfy a given mathematical equation or formula. Normally, if you insert a solution into an equation, it should make the equation true.

Step 2: Identifying Non-Satisfying Solutions

Sometimes, certain types of solutions do not satisfy their original equations. One such type is an 'extraneous' solution. An extraneous solution is an apparent solution that must be rejected because it does not satisfy the original equation after verification.

Step 3: Defining the Answer

Considering the question and the nature of solutions, we can tell that the blank is referring to an 'extraneous' solution because an extraneous solution is the type of solution that does not satisfy the original equation.

Key Concepts

Mathematical Solutions

In mathematics, when we solve an equation, we are looking for the 'solution.' A solution is rather straightforward to understand - it is a number or a set of numbers that, when substituted back into the equation, make the equation true.
For example, if we have an equation like \(x + 2 = 5\), the solution would be \(x = 3\), because substituting 3 for x renders the equation true as \(3 + 2 = 5\).

• A mathematical solution should satisfy the equation.
• If inserted back into the original equation, it should keep both sides of the equation equal.

It's like solving a puzzle, where only certain pieces (solutions) fit perfectly into the equation's framework.

Equation Verification

Equation verification is a crucial step in validating solutions. Once you have determined potential solutions, it's important to verify them. Verification involves substituting the solution back into the original equation to ensure it balances.
This step helps in identifying solutions that do not actually satisfy the equation, known as extraneous solutions. These can arise during manipulation of the equation (such as squaring both sides) and must be checked.

• Always plug back the solution into the original equation.
• Ensure the equation holds true - both sides should equal.

Skipping this step might lead to accepting solutions that aren't truly viable.

Types of Solutions

Understanding the types of solutions is important in solving equations. Here are the common ones:

• True Solutions: These satisfy the original equation and are valid solution sets.
• Extraneous Solutions: These appear as potential solutions during calculations but do not satisfy the original equation upon verification.

Extraneous solutions often surface in equations involving squares, roots, or rational expressions. For instance, when both sides of a simple equation are squared, it might introduce solutions that seem valid mathematically, but don't hold up upon verification.
Knowing these types helps avoid errors and ensures you determine correct solutions efficiently.