Question: If $ 2a + 3b = 14 $ and $ a - b = 1 $, what is the value of $ 5a + 2b $?
["Title: Solve for $ a $ and $ b $: A Step-by-Step Guide to $ 5a + 2b $ Given $ 2a + 3b = 14 $ and $ a - b = 1 $", "---", "Introduction\nIf you’re tackling systems of equations, you’re not alone—many students and self-learners face similar challenges. One common question is: If $ 2a + 3b = 14 $ and $ a - b = 1 $, what is the value of $ 5a + 2b $? This article walks you through solving this problem step-by-step, using standard algebraic methods. By the end, you’ll not only find $ 5a + 2b $, but also understand how to confidently solve similar systems in the future.", "---", "Step 1: Understand the Given Equations\nWe are given two linear equations:\n1. $ 2a + 3b = 14 $ (Equation 1)\n2. $ a - b = 1 $ (Equation 2)", "Our goal is to determine the value of $ 5a + 2b $. The key is to first solve for $ a $ and $ b $, then substitute or combine values to compute $ 5a + 2b $.", "---", "Step 2: Solve Equation 2 for One Variable\nStart with Equation 2:\n$$\na - b = 1\n$$\nSolve for $ a $:\n$$\na = b + 1\n$$", "---", "Step 3: Substitute into Equation 1\nNow substitute $ a = b + 1 $ into Equation 1:\n$$\n2(b + 1) + 3b = 14\n$$\nExpand:\n$$\n2b + 2 + 3b = 14\n$$\nCombine like terms:\n$$\n5b + 2 = 14\n$$\nSubtract 2 from both sides:\n$$\n5b = 12\n$$\nDivide by 5:\n$$\nb = \frac{12}{5}\n$$", "---", "Step 4: Find the Value of $ a $\nRecall $ a = b + 1 $. Substitute $ b = \frac{12}{5} $:\n$$\na = \frac{12}{5} + 1 = \frac{12}{5} + \frac{5}{5} = \frac{17}{5}\n$$", "---", "Step 5: Compute $ 5a + 2b $\nNow calculate:\n$$\n5a + 2b = 5\left(\frac{17}{5}\right) + 2\left(\frac{12}{5}\right) = 17 + \frac{24}{5}\n$$\nConvert 17 to fifths:\n$$\n17 = \frac{85}{5}\n$$\nSo:\n$$\n\frac{85}{5} + \frac{24}{5} = \frac{109}{5}\n$$", "---", "Final Answer:\n$$\n\boxed{\frac{109}{5}} \quad \ ext{or} \quad \boxed{21.8}\n$$", "---", "Conclusion\nUsing substitution with one equation and basic algebraic operations, we solved for $ a $ and $ b $, then computed $ 5a + 2b $. This method works reliably for any system of two linear equations. Practice helps build speed and confidence—so next time you face $ 2a + 3b = 14 $ and $ a - b = 1 $, chase the solution with clarity and precision!", "---", "Keywords: solve linear equations, systems of equations, algebra practice, solve for $ a $ and $ b $, $ 5a + 2b", step-by-step math tutorial, equation substitution, algebra homework help", "Meta Description:\nWant to know what is $ 5a + 2b $ if $ 2a + 3b = 14 $ and $ a - b = 1 $? This step-by-step guide walks you through solving the system and finding the exact value."]
