Now subtract eq1 from eq2: \( (6a + b) - (3a + b) = 6 - (-1) \Rightarrow 3a = 7 \Rightarrow a = \frac{7}{3} \)

["# How to Solve Linear Equations Like ( (6a + b) - (3a + b) = 6 - (-1) \Rightarrow 3a = 7 \Rightarrow a = \frac{7}{3} )", "Solving linear equations is a fundamental skill in algebra, essential for students, educators, and anyone working in math-sensitive fields. One powerful method involves carefully simplifying expressions by combining like terms and isolating variables. Consider the equation:", "[\n(6a + b) - (3a + b) = 6 - (-1)\n]", "In this step-by-step guide, we’ll break down how to solve this equation and derive the value of ( a ), resulting in ( a = \frac{7}{3} ). Whether you're preparing for a test, teaching algebra, or solving real-world problems, mastering this technique strengthens your algebraic fluency.", "## Understanding the Equation Structure", "Start by recognizing what’s happening in the equation:", "- The left-hand side involves two expressions: ( (6a + b) ) and ( (3a + b) ), subtracted from one another.\n- The right-hand side simplifies to a constant: ( 6 - (-1) = 6 + 1 = 7 ).", "Our goal is to simplify the left side and isolate the variable ( a ).", "## Step 1: Simplify Both Sides Using Algebraic Rules", "### Simplify the Left-Hand Side\nUse the distributive property to expand the subtraction:", "[\n(6a + b) - (3a + b) = 6a + b - 3a - b\n]", "Notice that ( +b - b ) cancels out:", "[\n6a - 3a + b - b = 3a\n]", "So, the equation simplifies to:", "[\n3a = 7\n]", "This elimination of ( b ) astutely occurs because ( b ) appears with opposite signs and thus cancels out—protecting against common student errors.", "### Simplify the Right-Hand Side\nAs mentioned earlier:", "[\n6 - (-1) = 6 + 1 = 7\n]", "### Final Simplified Equation\nAfter simplification, we have:", "[\n3a = 7\n]", "## Step 2: Isolate the Variable ( a )", "Divide both sides by 3 to solve for ( a ):", "[\na = \frac{7}{3}\n]", "This clean, precise result shows how isolating variables through operations on both sides preserves equation balance.", "## Why Understanding Cancellation Matters", "The cancellation of ( b ) in ( b - b ) is critical. Many learners mistakenly think terms should individually simplify before broader operations. However, simplifying after expansion ensures accuracy—especially when signs differ. Recognizing when like terms combine (or cancel) is key to solving complex equations efficiently.", "## More Tips for Solving Similar Equations", "- Always simplify both sides first before solving for variables. Expand and combine like terms fully.\n- Use parentheses carefully to clarify operations and prevent sign errors.\n- Isolate one variable at a time using inverse operations—add, subtract, multiply, or divide both sides equally.\n- Double-check your work by restoring all terms and substituting the solution back.", "For example, verify:\nPlug ( a = \frac{7}{3} ) into original equation:", "Left: ( (6 \cdot \frac{7}{3} + b) - (3 \cdot \frac{7}{3} + b) = (14 + b) - (7 + b) = 7 )\nRight: ( 7 ) → Equation balanced.", "## Conclusion", "Solving equations like ( (6a + b) - (3a + b) = 6 - (-1) ) is about strategic simplification and careful algebra. By systematically expanding, combining like terms, canceling variables when possible, and isolating ( a ), you transform expressions into clear, solvable form. Mastering these steps empowers smooth navigation of algebra—turning equations into answers.", "Key Takeaways:\n- Cancel ( b ) terms when they appear with opposite signs\n- Simplify both sides fully before solving for the unknown\n- Always isolate variables using inverse operations\n- Verify solutions by substitution", "Whether tackling homework, qualifying for exams, or analyzing data, algebra clarity starts with these foundational skills.\nStart practicing — each equation sharpens your logic and confidence!"]