Problem 12 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the equation is an identity, a conditional equation, or a contradiction. $$-5(x-1)=-5(x+1)$$

Step-by-Step Solution

Answer

The given equation is a contradiction.

Step 1: Distribute

First, distribute the -5 on both sides of the equation. This will give: \(-5 \cdot x + 5 = -5 \cdot x -5 \)

Step 2: Simplify

Now that we have removed the parenthesis by distributing, the equation simplifies to: \(-5x + 5= -5x -5\)

Step 3: Compare both sides

Finally, comparing both sides, it is clear that they are not equal to each other. Thus, the equation is neither an identity nor a conditional equation, making it a contradiction.

Key Concepts

Conditional Equation

A conditional equation is a type of mathematical statement that is true only for certain values of its variables. These equations will not hold true under all possible conditions; instead, they depend on specific conditions being met.

For example, consider the equation \( 2x + 3 = 11 \). This is a conditional equation, as it is only true when \( x \) is equal to 4. In such cases, the solution to the equation is the particular value, or set of values, that satisfy the equation when substituted into it.

These equations are incredibly common in algebra and often require the student to perform a series of operations to isolate the variable, providing a clearer understanding of for which values the equation holds true. Simplification and manipulation techniques are vital for solving conditional equations, which lead us towards the simplifying of equations.

Contradiction

In mathematics, a contradiction is a type of equation or statement that is false for all values of the variables involved. This is in sharp contrast to an identity, which is true for all values, and conditional equations, which are true for specific values.

Let's consider the equation from the exercise: \( -5(x-1)=-5(x+1) \). After simplifying, we get \( -5x + 5 = -5x -5 \) which implies that \( 5 = -5 \), which is clearly not true for any real number. Therefore, this equation is classified as a contradiction. Contradictions might initially appear to be correctly structured equations, but upon closer inspection, they prove to have no solution.

Identifying contradictions is important as it prevents students from wasting time trying to solve an unsolvable equation. Recognizing a contradiction early in the problem-solving process is a valuable skill in algebra and higher mathematics.

Simplifying Equations

The process of simplifying equations involves performing operations to rewrite equations in a more basic or more easily understandable form without changing the equation's solutions. This step is vital when solving equations as it helps to clearly identify what the equation represents; whether it's a conditional equation, an identity, or a contradiction.

To simplify an equation, you may combine like terms, use the distributive property, cancel out terms on both sides, or factor expressions. For instance, in our exercise, the equation \( -5(x-1)=-5(x+1) \) was initially simplified by distributing the \( -5 \) into the parenthesis which gave us a simpler form: \( -5x + 5 = -5x -5 \). This simplified form made it easier to see that both sides were not equal, leading to the conclusion that it was a contradiction.

Simplifying equations is not only a step towards solving an equation but also a tool for better understanding the structure of the equation, which is crucial for mathematical comprehension and problem-solving abilities.