Problem 45 - 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. $$ \frac{t+1}{2}=\frac{3 t-2}{6} $$

Step-by-Step Solution

Answer

The equation has no solutions and is a contradiction.

Step 1: Clear the Denominators

First, get rid of the fractions by multiplying both sides of the equation by a common multiple of the denominators, which in this case is 6, to simplify the fractions. This gives us: \[ 6 \cdot \frac{t+1}{2} = 6 \cdot \frac{3t-2}{6} \]This simplifies to:\[ 3(t+1) = 3t-2 \]

Step 2: Expand and simplify the equation

Distribute the terms on the left-hand side:\[ 3(t+1) = 3t + 3 \]Substitute back into the equation:\[ 3t + 3 = 3t - 2 \]

Step 3: Isolate the variable

To solve for \( t \), first subtract \( 3t \) from both sides to try to isolate the constant terms:\[ 3t + 3 - 3t = 3t - 2 - 3t \]This simplifies to:\[ 3 = -2 \]

Step 4: Analyze the result

The result \( 3 = -2 \) is clearly a false statement, indicating that our initial equation has no solution. Therefore, the initial equation is a contradiction.

Key Concepts

Solve Equation Symbolically

Solving an equation symbolically involves using algebraic manipulations to find the values of variables within the problem. It doesn't rely on graphical or numerical methods but instead on algebraic techniques, such as addition, subtraction, multiplication, division, and distribution. When solving symbolically, follow these general steps to arrive at the solution that satisfies the equation:

• Clear the Fractions: Multiply through by a common multiple of the denominators to eliminate fractions and make the equation easier to handle.
• Expand and Simplify: Distribute any terms and combine like terms to simplify the equation further.
• Isolate the Variable: Rearrange the equation to get the term containing the variable by itself on one side of the equation.
• Solve for the Variable: Perform the necessary operations to solve for the variable, ensuring you maintain balance in the equation.

In this particular exercise, we multiplied both sides by 6 to clear the fractions, which simplified our task to solving a simpler linear equation.

Classify Equation Types

Equations can be classified into different types based on the characteristics of their solutions:

• Contradiction: An equation with no solution because it results in an impossible statement, such as 3 = -2, as shown in the original solution.
• Identity: An equation that holds true for all possible values of the variable(s), such as 0 = 0. These equations describe equivalent expressions.
• Conditional Equation: An equation that is true for some specific values of the variable(s), such as 2x = 4, which is true when x = 2.

Understanding the type of equation you're dealing with helps determine the possible solutions and the methods suitable for handling them. In this case, we determined the equation is a contradiction because after isolating terms, we ended up with the false statement 3 = -2.

Equation Contradiction

A contradiction in an equation indicates there are no possible values for the variable(s) that make the equation true. Here, after simplifying the given equation \[ \frac{t+1}{2} = \frac{3t-2}{6} \]we eventually arrived at the statement 3 = -2, which is a contradiction. This happens when there is inherent inconsistency in the setup of the equation or initial conditions. Discovering a contradiction means that an assumption might be incorrect or the problem constraints are not compatible. Identifying contradictions is a crucial part of validating the consistency of any algebraic system.To avoid misinterpretations:

• Ensure each step in simplification is accurate.
• Reassess the problem constraints if a contradiction appears.
• Look out for mistakes or assumptions in the initial equation setup.