Problem 39 - College Algebra with Modeling and Visualization

Question

Exercises \(39-48:\) Complete the following. (a) Solve the equation symbolically. (b) Classify the equation as a contradiction, an identity, or a conditional equation. $$ 5 x-1=5 x+4 $$

Step-by-Step Solution

Answer

The equation is a contradiction.

Step 1: Set Up the Equation

Start with the given equation: \[ 5x - 1 = 5x + 4 \] Our goal is to solve this equation for \( x \).

Step 2: Eliminate the Variable on One Side

Subtract \( 5x \) from both sides of the equation to try to isolate terms: \[ 5x - 1 - 5x = 5x + 4 - 5x \] This simplifies to: \[ -1 = 4 \].

Step 3: Analyze the Result

Now, we have the statement \(-1 = 4\), which is false. This indicates that there are no values of \( x \) that can satisfy this equation. Thus, the given equation is a contradiction.

Key Concepts

Solve Equation

Solving an equation is like solving a puzzle where our mission is to find the right value for an unknown variable. In algebra, we often encounter equations that include an unknown represented by symbols, typically noted as "x". The main goal is to figure out what "x" should be, so that both sides of the equation are equal.

Let's take a closer look at the original problem:

• Given equation: \( 5x - 1 = 5x + 4 \)

To attempt solving this, we aim to isolate the variable \( x \) on one side of the equation. You'll often achieve this by adding, subtracting, multiplying, or dividing terms, balancing actions on both sides to preserve equality. Each step should bring you closer to a clearer understanding of what the equation is asking.

When handling equations, it's crucial to perform the same operation on both sides to maintain the equation's balance. This keeps both sides equal as you work to reveal the value of the unknown variable.

Contradiction in Equations

In algebra, not all equations are created equal. Some lead you to a truth, others present an unavoidable contradiction. A contradiction happens when simplifying an equation results in a false statement, such as \(-1 = 4\). This tells us that no value for the variable will ever satisfy the equation, meaning it's impossible to make both sides equal.

In our given equation, subtracting \(5x\) from both sides directly simplifies to \(-1 = 4\). This result is the perfect example of contradiction, since there's no number that can make \(-1\) equal to \(4\).

When faced with a result that is a contradiction, it means the original equation was set up in such a way that it cannot be solved for any real number of the variable. Recognizing contradictions saves time and helps determine the nature of the equations you're working with.

Symbolic Manipulation

Symbolic manipulation is a powerful tool in algebra, allowing you to transform and simplify equations to reveal insights. It's about cleverly using algebraic operations to move expressions around and make sense of complex problems.

In our example, we used symbolic manipulation to simplify the equation by subtracting \(5x\) from both sides:

• Subtracting terms helps eliminate variables, reducing equations to simpler forms.

With careful manipulation, we're left with an equation \(-1 = 4\). This false statement emerged from correct manipulative steps, proving an inherent contradiction in the setup.

Learning symbolic manipulation helps in various scenarios, enabling you to solve, simplify, or even determine the nature of an equation. Mastery of this skill is key to efficiently working through algebraic problems, emphasizing the importance of understanding the equality's balance.