Problem 46 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Solve the equation symbolically. (b) Classify the equation as a contradiction, an identity, or a conditional equation. $$ \frac{2 x+1}{3}=\frac{2 x-1}{3} $$

Step-by-Step Solution

Answer

The equation has no solution; it is a contradiction.

Step 1: Set Up the Equation

Start with the given equation: \( \frac{2x+1}{3} = \frac{2x-1}{3} \). Observe that both sides have the same denominator.

Step 2: Eliminate the Denominator

Multiply both sides of the equation by 3 to eliminate the denominator: \[ 3 \times \frac{2x+1}{3} = 3 \times \frac{2x-1}{3} \] which simplifies to \[ 2x + 1 = 2x - 1. \]

Step 3: Simplify the Equation

Subtract \(2x\) from both sides of the equation:\[ 2x + 1 - 2x = 2x - 1 - 2x \] which simplifies to \[ 1 = -1. \]

Step 4: Analyze the Equation

The simplified equation \(1 = -1\) is a contradiction because 1 and -1 are not equal.

Step 5: Classify the Equation

Since the simplification resulted in a false statement, the equation is classified as a contradiction; it has no solution.

Key Concepts

Symbolic Solving

Symbolic solving involves manipulating equations using algebraic symbols to find solutions without necessarily interpreting them immediately.
By working with symbols, we perform mathematical operations to simplify and solve equations.

• First, observe the structure of the expression, and equalize factors by making sure that denominators are removed, as seen with multiplying by 3 in our example.
• Next, simplify further by using basic algebraic operations such as addition, subtraction, multiplication, or division.
• Keep track of equivalent transformations to ensure the expressions are simplified correctly.

After solving symbolically, we analyze the form of the resulting equation, which brings us closer to understanding if the equation is solvable or not.

Conditional Equations

A conditional equation is an equation that is true only for specific values of its variable. These equations have at least one solution that satisfies the equation.
When solving such equations, our objective is to isolate the variable and find values that make the equation true, which involves steps like:

• Combining like terms and removing constants as necessary.
• Using inverse operations to isolate the variable.
• Verifying the potential solution by substituting it back into the original equation to ensure its truth.

In our specific exercise, after simplifying, the final result unfortunately wasn't conditional, but this process highlights the necessary steps one would take if it had been.

Identity Equations

Identity equations are equations that are true for all variable values. No matter what number you substitute for the unknown variable, the equation holds true.
Key features of identity equations include:

• The algebraic sentence balances itself perfectly during simplification.
• The simplification leads to a tautology such as "0 = 0" or "5 = 5."
• They imply infinite solutions, as any number will satisfy the equation.

Unlike our initial problem, where a contradiction arose, identity equations represent an interesting case of universal truth within algebra.