- 6b + 25b = 155 \Rightarrow 19b = 69 \Rightarrow b = \frac{69}{19}

["Solving the Equation: 6b + 25b = 155 – A Step-by-Step Breakdown", "When solving linear equations, understanding each step clearly helps reinforce mathematical logic and strengthens problem-solving skills. One common type is the simple linear equation involving a variable multiplied by coefficients. Today, we explore the equation 6b + 25b = 155, walk through its solution step-by-step, and arrive at b = $\frac{69}{19}$.", "---", "### Understanding the Equation", "The equation starts as:", "$$\n6b + 25b = 155\n$$", "Here, b is the variable we need to solve for. Both 6b and 25b are terms containing the same variable, so we combine like terms:", "### Step 1: Combine Like Terms", "Add the coefficients of b:", "$$\n(6 + 25)b = 155 \Rightarrow 31b = 155\n$$", "Wait — this result, 31b = 155, does not match the expected 19b = 69 stated in the prompt. Let’s verify carefully, because the problem statement says:", "$$\n6b + 25b = 155 \Rightarrow 19b = 69 \Rightarrow b = \frac{69}{19}\n$$", "This discrepancy indicates a possible typo in the original equation. Let's re-express the correct and consistent derivation.", "---", "### Correcting the Equation to Match the Expected Result", "To align with b = $\frac{69}{19}$, let’s assume the original equation was intended to be:", "$$\n19b = 69\n$$", "From this, solving for b is straightforward:", "### Step 1: Isolate b", "Divide both sides by 19:", "$$\nb = \frac{69}{19}\n$$", "This fraction simplifies to:", "$$\nb = 3.631578\ldots \quad \ ext{(approximate decimal)}\n$$", "But for exact form, $ \frac{69}{19} $ remains as is.", "---", "### Why Does This Equation Equal 19b = 69?", "If the original equation is corrected to:", "$$\n19b = 69\n$$", "it directly reflects a single step of simplification from 6b + 25b = 155 only if additional assumptions or alternate operations were applied. However, as shown, 6b + 25b = 31b, not 19b.", "Conclusion:\n- The equation 6b + 25b = 155 simplifies to 31b = 155 with solution b = 5.\n- The derived equation 19b = 69 and solution b = $\frac{69}{19}$ originates from a different equation.", "---", "### Real Scenario: Applied Math Behind the Equation", "Suppose in a real-world context, we model cost or growth with combined rates:", "- Let b represent a base rate (e.g., per unit time, per component).\n- If each unit contributes 6b and 25b collectively sum to 155 units, yet mistakenly simplified aspects lead to 19b = 69, perhaps from misaligned coefficients or transcription.", "Nonetheless, to honor your prompt:", "---", "### Final Solution Restated Clearly", "From 6b + 25b = 155:", "1. Combine coefficients:\n $$\n 31b = 155\n $$\n2. Solve for b:\n $$\n b = \frac{155}{31} = 5\n $$", "To satisfy the stated result b = $\frac{69}{19}$, the original equation must be altered — for instance:", "$$\n(6 + 13)b = 155 \Rightarrow 19b = 155 \Rightarrow b = \frac{155}{19}\n$$", "Close, but still mismatched. The only coherent match is:", "> Corrected Equation: 19b = 69 → b = $\frac{69}{19}$", "Thus, if your equation was 6b + 13b = 155, then:", "$$\n19b = 155 \quad\ ext{(not 19b = 69)}\n$$", "Wait — this also fails.", "But if the intended equation was:", "$$\n(6 + 13)b = 69 \Rightarrow 19b = 69\n$$", "Then b = $\frac{69}{19}$ is correct.", "---", "### Summary: How to Work with Such Equations", "1. Accurately parse the equation — verify coefficients signs, operands.\n2. Combine like terms with care.\n3. Solve for variables using inverse operations.\n4. Double-check results and simplifications to avoid errors.\n5. Align equation structure with target variable solution for educational or problem-solving purposes.", "---", "### Why It Matters: Precision in Algebra", "Misstatements in equations can lead to incorrect solutions. Always verify each transformation:", "- Combining like terms is fundamental, but only when coefficients truly apply to the same variable.\n- View equations as logical puzzles where any mismatched terms break down the solution path.", "---", "### Final Thoughts", "While 6b + 25b = 155 leads logically to b = 5, the elegant outcome b = $\frac{69}{19}$ implies a different equation — likely 19b = 69. Whether through typo, simplification mistake, or algebraic reinterpretation, mastering the step-by-step process ensures accuracy and confidence in mathematics.", "---", "Keywords:\nHow to solve 6b + 25b = 155, solve linear equations, simplify combined terms, algebraic steps, b = 69/19 derivation, equity in math learning, solve for b algebraically, equation simplification guide", "Meta Description:\nLearn how to solve linear equations step-by-step. See how 6b + 25b = 155 simplifies to 31b = 155, and understand why the expected result b = $\frac{69}{19}$ comes from 19b = 69 — with accurate algebraic reasoning."]