\[ 3(2b - 1) + 4b = 12 \] - United Radiology

["# Solving ( 3(2b - 1) + 4b = 12 ): A Step-by-Step Guide", "If you’re working on simplifying linear equations, mastering the ability to solve expressions like ( 3(2b - 1) + 4b = 12 ) is essential. Whether you're a student, teacher, or self-learner, understanding how to approach this equation not only improves your math skills but also strengthens your algebraic reasoning. In this article, we break down step-by-step how to solve ( 3(2b - 1) + 4b = 12 ), provide key strategies, and explain why each step matters.", "## Understanding the Equation", "The equation ( 3(2b - 1) + 4b = 12 ) combines arithmetic expressions involving variables, parentheses, and constants. Solving it requires applying the order of operations (PEMDAS/BODMAS), distributing terms, and combining like terms. Let’s walk through the process clearly.", "---", "## Step 1: Distribute the 3 Across the Parentheses", "Begin by eliminating the parentheses using the distributive property:", "[
\n3(2b - 1) = 3 \cdot 2b - 3 \cdot 1 = 6b - 3
\n]", "Substitute this back into the original equation:", "[
\n6b - 3 + 4b = 12
\n]", "---", "## Step 2: Combine Like Terms", "Now combine the terms with ( b ):", "[
\n(6b + 4b) - 3 = 12 \quad \Rightarrow \quad 10b - 3 = 12
\n]", "This step reduces complexity by grouping ( b )-terms together, making the equation easier to isolate ( b ).", "---", "## Step 3: Isolate the Variable Term", "Add 3 to both sides to eliminate the constant subtraction:", "[
\n10b - 3 + 3 = 12 + 3 \quad \Rightarrow \quad 10b = 15
\n]", "Isolating ( b ) simplifies solving for its value.", "---", "## Step 4: Solve for ( b )", "Divide both sides by 10:", "[
\nb = \frac{15}{10} = \frac{3}{2}
\n]", "This final step reveals the solution: ( b = \frac{3}{2} ), or 1.5.", "---", "## Additional Insights", "- Why distribute first? Distributing ensures that every term inside parentheses is accounted for, preserving equation balance.
\n- Importance of combining like terms: Combining like terms minimizes computation errors and streamlines the solution.
\n- Checking the solution: Substitute ( b = \frac{3}{2} ) back into the original equation:", "[
\n3\left(2 \cdot \frac{3}{2} - 1\right) + 4 \cdot \frac{3}{2} = 3(3 - 1) + 6 = 3 \cdot 2 + 6 = 6 + 6 = 12
\n]", "The left side equals the right, confirming correctness.", "---", "## Why Learn This Problem?", "Solving linear equations like ( 3(2b - 1) + 4b = 12 ) is not just an academic exercise—it builds foundational skills used in science, engineering, economics, and everyday problem-solving. These equations model real-world relationships, from budgeting to physics formulas.", "---", "## Summary", "To solve ( 3(2b - 1) + 4b = 12 ), follow these key steps:
\n1. Distribute: ( 3(2b - 1) \rightarrow 6b - 3 )
\n2. Combine like terms: ( 10b - 3 )
\n3. Isolate ( b ): Add 3, then divide by 10
\n4. Verify by substituting back", "Final Answer:
\n[
\n\boxed{b = \frac{3}{2}}
\n]", "Master this process confidently—your algebra skills will grow stronger with every equation solved!"]