Problem 48 - Fundamentals of Mathematics

Question

Calculator Exercises. $$4.23 k+3.18=3.23 k-5.83$$

Step-by-Step Solution

Answer

The solution is \(k = -9.01\).

Step 1: Isolate the variable terms

First, we want to gather all the terms with the variable on one side of the equation and the constant terms on the other. Subtract \(3.23k\) from both sides: \[4.23k - 3.23k + 3.18 = -5.83\].

Step 2: Simplify the equation

Now, simplify the left side of the equation by combining the \(k\) terms: \[(4.23 - 3.23)k + 3.18 = -5.83\] which simplifies to: \[1k + 3.18 = -5.83\].

Step 3: Isolate the variable

To solve for \(k\), subtract \(3.18\) from both sides of the equation:\[1k = -5.83 - 3.18\] which simplifies to: \[k = -9.01\].

Step 4: Final Check

Finally, substitute \(k = -9.01\) back into the original equation to ensure the solution is correct:\[4.23(-9.01) + 3.18 = 3.23(-9.01) - 5.83\] which simplifies to both sides being approximately equal. Hence, the solution is verified.

Key Concepts

Variable Isolation

When solving linear equations, one of the key steps is **variable isolation**. In this context, 'isolating the variable' means getting the variable you're trying to solve for, which is usually represented by a symbol like \(k\), by itself on one side of the equation. This allows us to find its exact value. Let's see how this works in practice using the equation: \(4.23k + 3.18 = 3.23k - 5.83\).

- **Start by separating variable terms**: In our equation, the terms involving \(k\) are on both sides. To isolate \(k\), I first subtract \(3.23k\) from both sides. This approach helps in gathering all terms with \(k\) on one side. Here's how the equation adjusts:
- New equation: \(4.23k - 3.23k + 3.18 = -5.83\).

Variable isolation is all about systematic adjustments to ensure all variable terms are consolidated. It sets a strong foundation for the next step: simplification.

Simplification of Equations

After isolating the variable, **simplification of equations** follows naturally. Simplifying means making the equation as straightforward as possible, which typically involves combining like terms and simplifying expressions. Let's explore this concept using our current equation:
- We have now: \(4.23k - 3.23k + 3.18 = -5.83\).
- **Combine the \(k\) terms**: Here, the \(k\) terms are on the left side and can be combined. Calculate: \(4.23 - 3.23\), which simplifies the left side to:
- \(1k + 3.18 = -5.83\).

Simplification through combining like terms and reducing unnecessary components helps in viewing the equation clearly. With each simplification step, you bring the problem closer to the solution.

In this case, the simplified form directly helps us move towards finding \(k\).

Verification of Solutions

Once you solve an equation and find a solution, the last key step is **verification of solutions**. Verifying means making sure that the answer you've found actually satisfies the original equation. This process is necessary for confirming accuracy. Here's how to verify using our solution, \(k = -9.01\):
**Substitute the result back into the original equation**:
- Original equation: \(4.23k + 3.18 = 3.23k - 5.83\).
- Replace \(k\) with \(-9.01\) and check both sides:
- Left side: \(4.23(-9.01) + 3.18\).
- Right side: \(3.23(-9.01) - 5.83\).
- **Compare the results**: Both sides should evaluate to approximately the same value if the solution is correct.

Through substitution and equal comparison, verification ensures that no mistakes were made during calculations. This step is crucial not only for self-confidence in one's solution but also for mastering linear equations with precision.