Subtract (2) from (3): $ 5a + b = 900 $ (5)

["Understanding the Subtraction of (2) from (3): Solving the Equation $ 5a + b = 900 $ (5)", "In algebra, solving equations step by step is essential for clarity and accuracy. One common operation is subtracting one equation from another, commonly referred to in textbook problems as subtracting (2) from (3). In this article, we explore how to effectively subtract equation (2) from equation (3) using the linear equation $ 5a + b = 900 $ (5) as a guiding example.", "### What Does “Subtract (2) from (3)” Mean?", "The phrase “subtract (2) from (3)” typically means performing an algebraic operation where one equation is subtracted entirely from another. This technique helps eliminate a variable, uncover unknown values, and simplify complex systems of equations.", "Even though equation (5) is given as $ 5a + b = 900 $, imagine this equation is labeled (5), and we are comparing or combining it with another equation—say (2)—to isolate variables and solve the system. Subtracting equation (2) from (3) means:", "$$\n(5a + b) - (5a + b) = 900 - (??)\n$$", "But to truly understand subtraction in this context, we must assume (2) represents a related linear equation, such as $ 5a - 2b = x $, to make a meaningful subtraction.", "### Step-by-Step: Subtracting Equation (2) from (3)", "Let’s assume equation (2) is:\n$$\n5a - 2b = 400 \quad \ ext{(2)}\n$$\nand equation (3) is:\n$$\n5a + b = 900 \quad \ ext{(5)}\n$$", "Now subtract equation (2) from equation (3):", "$$\n(5a + b) - (5a - 2b) = 900 - 400\n$$", "Distribute the negative sign:", "$$\n5a + b - 5a + 2b = 500\n$$", "Combine like terms:", "$$\n(5a - 5a) + (b + 2b) = 500\n\Rightarrow 0a + 3b = 500\n\Rightarrow 3b = 500\n$$", "Solve for $ b $:\n$$\nb = \frac{500}{3} \approx 166.67\n$$", "Now substitute $ b $ back into equation (5):", "$$\n5a + \frac{500}{3} = 900\n\Rightarrow 5a = 900 - \frac{500}{3} = \frac{2700 - 500}{3} = \frac{2200}{3}\n\Rightarrow a = \frac{2200}{15} = \frac{440}{3} \approx 146.67\n$$", "### Why Subtract (2) from (3)?", "Subtracting equations strategically allows elimination of variables and simplification. Here, by subtracting (2) from (3), we removed the $ 5a $ term since $ 5a - 5a = 0 $, making it easier to solve for $ b $. This step is fundamental in methods like elimination, substitution, and solving systems of equations.", "### Summary", "- Subtracting one equation from another (like (2) from (3)) simplifies solving linear systems.\n- Always combine like terms after subtracting to reduce the equation to a single variable.\n- In our example, subtracting $ 5a - 2b $ from $ 5a + b $ successfully eliminated $ a $.\n- Solving yields $ a = \frac{440}{3} $, $ b = \frac{500}{3} $.", "### Key Takeaways", "- Use equation subtraction to eliminate variables efficiently.\n- Carefully combine like terms after subtraction.\n- Always substitute back to find full solutions.\n- Practice with real equations to master this algebraic technique.", "Mastering equation subtraction strengthens your ability to solve multivariable problems — a crucial skill in algebra, calculus, and beyond.", "---", "Keywords:\nsubtract equation (2) from (3), solving linear equations, algebra guide, elimination method, linear systems, solving for variables, $ 5a + b = 900 $, equation subtraction technique."]