Problem 77 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. (Some of the equations have no solution.) $$2 s+\frac{3}{2}=2 s+2$$

Step-by-Step Solution

Answer

The equation \(2s + \frac{3}{2} = 2s + 2\) has no solution.

Step 1: Equation Simplification

First, subtract \(2s\) from both sides of the equation to simplify the equation. This will give us \( \frac{3}{2} = 2\).

Step 2: Evaluating and Solving

Upon evaluating \( \frac{3}{2} = 2\), we can see that the statement on its own is not true, since \(1.5\) does not equal \(2\). So in this case, we cannot find a value for \(s\) that satisfies the equation.

Step 3: Concluding the Solution

Since the equation does not hold true, it means that there is no solution for \(s\). In other words, there does not exist an \(s\) which can make both sides of the equation equal.

Key Concepts

Simplification Techniques

When solving equations, simplification is a vital process that streamlines complex expressions into a more easily manageable form. This involves removing unnecessary elements or terms, which can make solving equations far simpler. In our example, the equation given is:

• \( 2s + \frac{3}{2} = 2s + 2 \)

The first step in simplification is to eliminate terms that appear on both sides of the equation. This can be done by subtracting identical terms from each side. Here, by subtracting \( 2s \) from both parts, we drastically reduce the equation's complexity:

• Resulting in: \( \frac{3}{2} = 2 \)

This action helps us focus on what's actually necessary to solve the equation. Simplification methods like these are pivotal in achieving clarity in mathematical problem-solving.

No Solution Concept

Sometimes, after simplifying an equation, you might find that no possible value can satisfy the equation. This is known as the "no solution" scenario. In our example, after simplification, we ended with the statement:

• \( \frac{3}{2} = 2 \)

This equation is inherently false, as \(1.5\) (the decimal value of \(\frac{3}{2}\)) is not equal to 2. This type of outcome indicates there is no possible number, known as "no solution," that could make the original equation true.
In practical terms, encountering such a result means there are no real-world values or instances where the equation can be satisfied. Understanding this concept is crucial, as it saves time by recognizing impossibility in solutions early on.

Equation Evaluation

Evaluating an equation involves assessing the simplified form to determine if there is a feasible solution. After streamlining, questions remain:

• Does a number exist that makes both sides equal?

In our simplification example resulting in \( \frac{3}{2} = 2 \), evaluation quickly elucidates the impossibility: there simply isn't a number that can equate 1.5 and 2. For practical application, this step verifies the work done during simplification and confirms assumptions.
In summary, effective equation evaluation aids in quickly identifying "no solution" cases, ensuring time and resources are spent only when solutions truly exist. This highlights the importance of thoroughly checking work, bolstered by rigorous logic and a foundational understanding of mathematical principles.