Verified Answer :

  • Simplify the left side of the equation by dividing each term by 3, which results in \(5x + 2\).
  • Equate this simplified expression to the right side, giving \(5x + 2 = 5x - 2\).
  • Observe that subtracting \(5x\) from both sides leads to the contradiction \(2 = -2\).
  • Conclude that there is no solution to the equation.
  • Final Answer: \(\boxed{\text{no solution}}\)

Explanation

  1. Simplify and Analyze the Equation

The objective is to find the value of \(x\) that satisfies the equation \(\frac{15x+6}{3} = 5x - 2\). We begin by simplifying the left side of the equation. We can distribute the division by 3 across the terms in the numerator:
\[\frac{15x}{3} + \frac{6}{3} = 5x + 2.\]
Now, substitute this back into the original equation:
\[5x + 2 = 5x - 2.\]

  1. Conclusion about the Solution

To determine the solution, we attempt to isolate \(x\). If we subtract \(5x\) from both sides of the equation, we get:
\[5x + 2 - 5x = 5x - 2 - 5x\]
\[2 = -2.\]
Since the resulting statement \(2 = -2\) is a mathematical contradiction (a statement that is never true), there is no value of \(x\) that can satisfy the original equation. Therefore, we conclude that the equation has no solution.

Examples

Imagine you are trying to calculate the amount of fuel in two different tanks. Tank A starts with 6 units and gains fuel at a rate of 5 units per hour, represented by \(5x + 2\), while Tank B starts at -2 units and gains fuel at the same rate, represented by \(5x - 2\). If you want to know when both tanks have the same amount of fuel, you set the expressions equal. Since the tanks gain fuel at the exact same rate but started with different amounts, they will never have the same amount of fuel, reflecting that there is no solution to the equation.

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