Solving equations 1 and 2: From 1, $ a = 5 - 3b $. Substituting into 2: $ 2(5 - 3b) + 4b = 6 \Rightarrow 10 - 6b + 4b = 6 \Rightarrow -2b = -4 \Rightarrow b = 2 $. Then $ a = 5 - 6 = -1 $.
["Solving Linear Equations: A Step-by-Step Guide to Finding $ a $ and $ b $", "Solving equations is a fundamental skill in algebra, essential for everything from simple math problems to advanced engineering calculations. In this article, we’ll walk through a practical example: solving a system of linear equations using substitution. By carefully substituting and simplifying, we’ll find the values of $ a $ and $ b $ that satisfy both equations.", "---", "### The Problem at Hand", "We are given two equations:", "1. $ a = 5 - 3b $\n2. $ 2(5 - 3b) + 4b = 6 $", "Our goal is to solve for both $ a $ and $ b $. This system demonstrates how substitution simplifies solving for multiple variables.", "---", "### Step-by-Step Solution", "We start with the second equation, which contains the expression for $ a $ from the first equation.", "#### Step 1: Substitute Equation 1 into Equation 2", "Substitute $ a = 5 - 3b $ into the second equation:", "$$\n2(5 - 3b) + 4b = 6\n$$", "#### Step 2: Expand and Simplify", "Distribute the 2:", "$$\n10 - 6b + 4b = 6\n$$", "Combine like terms:", "$$\n10 - 2b = 6\n$$", "#### Step 3: Isolate $ b $", "Subtract 10 from both sides:", "$$\n-2b = 6 - 10\n$$\n$$\n-2b = -4\n$$", "Divide both sides by $-2$:", "$$\nb = 2\n$$", "---", "### Step 4: Solve for $ a $", "Now substitute $ b = 2 $ into the expression from equation 1:", "$$\na = 5 - 3b = 5 - 3(2) = 5 - 6 = -1\n$$", "---", "### Final Answer", "$$\na = -1,\quad b = 2\n$$", "---", "### Why This Method Works", "This approach shows how substitution connects the two equations into one, making it easier to solve. By replacing one variable using an already known expression, we reduce a system of two equations with two variables into a single simpler equation. This technique is powerful not just for equations like this, but for systems involving more complex relationships.", "---", "### Learning Takeaway", "Mastering substitution in equation solving builds confidence in algebra and prepares learners for more complex topics like graphing linear systems, solving word problems, and analytical reasoning. Practice these steps regularly to sharpen your problem-solving skills and boost your mathematical fluency.", "---", "Keywords: solve equations, substitution method, linear equations, algebra tutorial, solve for variables, step-by-step algebra, learn to solve equations, $ a $ and $ b $, solve $ a = 5 - 3b $, $ b = 2 $, $ a = -1 $", "---", "Need more help with solving equations? Explore our guides on substitution, elimination, and graphing linear systems to strengthen your algebra foundation!"]
