Problem 55 - College Algebra

Question

In Exercises \(51-58,\) determine whether each equation is an identity, a conditional equation, or an inconsistent equation. $$ 4 x+5 x=8 x $$

Step-by-Step Solution

Answer

The equation \(4x + 5x = 8x\) is a conditional equation.

Step 1: Simplify the Equation

Simplify the given equation by combining like terms on the left-hand side of the equation. This results in \(9x = 8x\).

Step 2: Rearrange the Equation

Subtract \(8x\) from both sides to move all terms involving the variable to one side of the equation: \(9x - 8x = 8x - 8x\), which simplifies to \(x = 0\).

Step 3: Determine the Nature of the Equation

Based on the rearranged equation, we see that \(x\) can only be \(0\) for the equation to hold true. Therefore, this is a conditional equation, since it only holds true for \(x = 0\) .

Key Concepts

Identity Equation

Equations can take various forms, and understanding these can greatly simplify solving them. One such form is an identity equation. An identity equation is true for all values of the variable involved. For example, the equation \( x + 3 = x + 3 \) holds no matter what value you substitute for \( x \). In this case, you can rely on the fact that both sides of the equation will always be equal.

• An identity does not depend on any specific value.
• It's like a universal truth for that particular equation format.
• Examples include \( 2(x+1) = 2x + 2 \), as both sides remain equal regardless of \( x \).

Recognizing identity equations can save you time, as you don't have to attempt to solve for a single solution. You simply validate that the structure of the equation itself is balanced and correct.

Inconsistent Equation

The counterpart to an identity equation is an inconsistent equation. An inconsistent equation is one that has no solution. This occurs when your equation simplifies to a false statement, such as \( x + 2 = x + 5 \). If you perform operations to isolate \( x \), you'll end up with a contradiction, for instance, \( 2 = 5 \). No value of \( x \) can ever make this equation true.

• Recognize inconsistencies when you end with a false statement.
• No values will satisfy an inconsistent equation, so the solution set is empty.
• You often spot these by simplifying one side and checking for logical contradictions.

When working with equations, identifying an inconsistency right off the bat tells you there are no answers to be found, helping you decide your approach early on.

Equation Simplification

Equation simplification is a critical skill in mathematics. It involves reducing equations to their simplest form. This process allows clearer insights into the nature of the equations—whether they are conditional, identity, or inconsistent.

Steps in Simplifying Equations

• Combine like terms: Start by grouping related terms on each side of the equation. For example, in the equation \( 4x + 5x \), these terms combine to form \( 9x \).
• Use basic arithmetic operations: Add, subtract, multiply, or divide as needed to simplify the equation.
• Rearrange terms to understand the structure better: Move terms to one side to see if the equation can be solved.

In the given exercise, simplifying \( 9x = 8x \) into \( x = 0 \) directly helps us discern it as a conditional equation, holding true under specific conditions. Simplicity lets us find solutions more easily and understand the nature of mathematical problems.