Expand: 3x - 12 + 2x = 5x + 6 - Portal da AcÃºstica

How to Solve the Equation: 3x - 12 + 2x = 5x + 6

Solve algebraic expressions like 3x - 12 + 2x = 5x + 6 with confidence—this step-by-step guide breaks down the process of expanding, simplifying, and isolating variables to find the value of x. Whether you're a student learning algebra or seeking a refresher, understanding how to expand and simplify equations is essential for success in math and related STEM fields.

Understanding the Context

Understanding the Equation

At its core, the equation 3x - 12 + 2x = 5x + 6 requires simplifying both sides using the distributive and combining like terms. Expanding means identifying any parentheses (though none are present here), rearranging terms, and combining variables and constants.

Step 1: Combine Like Terms on Each Side

Key Insights

Start with the left side:
3x + 2x - 12
This simplifies to:
5x - 12

Now the right side:
5x + 6 (already simplified)

Now rewrite the equation:
5x - 12 = 5x + 6

Step 2: Expand and Simplify

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📰 The sum \( S_n \) of the first \( n \) terms of a geometric sequence is \( S_n = a \frac{r^n - 1}{r - 1} \). 📰 Here, \( a = 3 \), \( r = 2 \), and \( n = 6 \). 📰 So, \( S_6 = 3 \frac{2^6 - 1}{2 - 1} = 3 \times (64 - 1) = 3 \times 63 = 189 \).

Final Thoughts

In this case, no expansion is needed beyond combining like terms. The equation now matches:
5x - 12 = 5x + 6

Subtract 5x from both sides:
-12 = 6

Step 3: Analyze and Interpret

This result, -12 = 6, is a contradiction—meaning there is no solution that satisfies the original equation. Both sides simplify to unlike constants, so no value of x can make the equation true.

Why This Matters (Real-World Application)

Understanding why equations have no solution helps in many areas, from optimizing business models to debugging scientific experiments. Recognizing when equations resolve to contradictions strengthens problem-solving skills in algebra and beyond.