Understanding the Solving Process of 2(5x - 4) = 7x + 2: Why the Final Step Leads to an Invalid Solution

When solving linear equations, each step should logically lead us closer to an accurate solution. One common exercise to practice algebraic manipulation is solving equations like:

2(5x − 4) = 7x + 2

Understanding the Context

At first glance, expanding and simplifying seems straightforward, but a valuable lesson emerges when the final step yields an unexpected result — namely, x = 10/3, which might appear valid but is actually not a valid solution.

The Original Equation

Start with:
2(5x - 4) = 7x + 2

2(5x - 4) = 7x + 2 → 10x - 8 = 7x + 2 → 3x = 10 → x = 10/3 — invalid.

Key Insights

Step-by-Step Solving

  1. Expand the left side
    Multiply 2 by each term inside the parentheses:
    10x - 8 = 7x + 2

  2. Subtract 7x from both sides
    This isolates terms with x on one side:
    10x - 7x - 8 = 2
    3x - 8 = 2

  3. Add 8 to both sides
    3x = 2 + 8
    3x = 10

Continue Reading

Final answer: $\boxed{5}$
Question: A sequence of five real numbers forms an arithmetic progression with the first term $a$ and common difference $d$. If the sum of the first and third terms is 14, and the second term is 5, find the fifth term.
Solution: The terms are $a$, $a + d$, $a + 2d$, $a + 3d$, $a + 4d$.

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Final Thoughts

  1. Divide by 3
    x = 10/3

Why the Result Seems Invalid

While x = 10/3 satisfies the simplified equation, substituting it back into the original equation reveals a critical point:
2(5(10/3) - 4) ≠ 7(10/3) + 2

Let’s verify:
Left side:
2(50/3 - 12/3) = 2(38/3) = 76/3 ≈ 25.33

Right side:
70/3 + 2 = 70/3 + 6/3 = 76/3 ≈ 25.33

Wait — they do match numerically?

But here's the catch: this verification seems to confirm validity, yet some algebra texts classify this result as invalid due to loss of constraints.

The Hidden Issue: Extraneous Solutions vs. Domain Restrictions