\( 2x + 3y = 100 \)
["# Mastering the Linear Equation: How to Solve ( 2x + 3y = 100 )", "Understanding linear equations is fundamental in algebra and plays a key role in diverse fields such as engineering, economics, and data science. One of the most common forms encountered is the equation ( 2x + 3y = 100 ). Whether you're solving optimization problems, analyzing budget constraints, or modeling relationships in real-world scenarios, mastering this simple linear equation can unlock deeper analytical skills. This article explores how to solve ( 2x + 3y = 100 ), interpret its meaning, and apply it in practical contexts.", "---", "## What is the Equation ( 2x + 3y = 100 )?", "The equation ( 2x + 3y = 100 ) is a linear Diophantine equation relating two variables: ( x ) and ( y ). Here, ( x ) and ( y ) represent variables that can take any real value (in continuous contexts) or integers (in discrete scenarios). The goal is to find pairs of values ((x, y)) that satisfy the equation.", "---", "## Basic Solution Approach: Express One Variable in Terms of the Other", "To solve ( 2x + 3y = 100 ), follow these steps:", "1. Isolate one variable\n It’s easiest to express ( y ) in terms of ( x ):\n [\n 3y = 100 - 2x\n ]\n [\n y = \frac{100 - 2x}{3}\n ]", "2. Interpret the solution\n For every value of ( x ) that makes ( 100 - 2x ) divisible by 3 (or simply for any real ( x ) in numerical contexts), ( y ) is determined. This means there are infinitely many real solutions to this equation.", "3. Graphical interpretation\n On a coordinate plane, ( 2x + 3y = 100 ) represents a straight line. The slope-intercept form is:\n [\n y = -\frac{2}{3}x + \frac{100}{3}\n ]\n This line intersects the axes at:\n - ( x )-intercept: When ( y = 0 ), ( x = 50 )\n - ( y )-intercept: When ( x = 0 ), ( y \approx 33.33 )", "---", "## Finding Integer Solutions (Diophantine Equations)", "If ( x ) and ( y ) are intended to be integers (e.g., in budgeting or counting problems), the equation becomes a Diophantine equation. We seek integer pairs ((x, y)) such that ( 2x + 3y = 100 ).", "### Step 1: Ensure solvability\nA linear Diophantine equation ( ax + by = c ) has integer solutions if and only if ( \gcd(a, b) ) divides ( c ).\nHere, ( \gcd(2, 3) = 1 ), and ( 1 ) divides ( 100 ), so integer solutions exist.", "### Step 2: Find a particular solution\nStart with simple values:\nLet ( x = 2 ):\n[\n2(2) + 3y = 100 \Rightarrow 4 + 3y = 100 \Rightarrow y = \frac{96}{3} = 32\n]\nSo, ((2, 32)) is one integer solution.", "### Step 3: General solution\nSince ( \gcd(2, 3) = 1 ), all integer solutions are given by:\n[\nx = 2 + 3k\n]\n[\ny = 32 - 2k\n]\nwhere ( k ) is any integer.", "### Example solutions:\n- ( k = 0 ): ( (2, 32) )\n- ( k = 1 ): ( (5, 30) )\n- ( k = -3 ): ( (-7, 38) )\n- ( k = 10 ): ( (32, 20) )", "These represent a sequence of solutions spanning all integers.", "---", "## Practical Applications of ( 2x + 3y = 100 )", "This simple equation models many real-world situations:", "### 1. Budget Constraints\nSuppose ( x ) = number of apples ($2 each), ( y ) = number of bananas ($3 each), and total spending is $100. The equation balances your budget.", "### 2. Resource Allocation\nA company uses two resources, costing $2 and $3 per unit, totaling 100 resource units. The equation helps analyze trade-offs.", "### 3. Nutrition and Diet Planning\nIf ( x ) represents grams of protein and ( y ) grams of fat, with protein costing $2 and fat $3 per gram, the equation constrains your meal’s nutrient profile within $100.", "### 4. Math and Algorithm Design\nLinear equations underpin algorithms in computer science, optimization, and machine learning—where ( 2x + 3y ) may represent cost, time, or resource weights.", "---", "## Visualizing the Solution Space", "Graphically, plotting ( y = \frac{100 - 2x}{3} ) reveals a continuous straight line. Integer solutions appear as lattice points (points with integer coordinates) on this line. This visualization aids understanding for students, engineers, and data analysts alike.", "---", "## Tips for Solving and Analyzing ( 2x + 3y = 100 )", "- Use substitution or elimination for systems involving this equation.\n- Plug values systematically: Test integers for ( x ) or parameters ( k ) in the general solution.\n- Explore inequalities: To find ranges, replace equality with ( 2x + 3y < 100 ) or ( > 100 ) to define feasible regions.\n- Applications extend to systems: Combine with other equations to model multi-variable constraints.", "---", "## Conclusion", "The equation ( 2x + 3y = 100 ) might seem simple, but it opens doors to understanding linear relationships, integer solutions, and real-world problem-solving. Whether you’re balancing budgets, designing systems, or exploring algebra, mastering this equation strengthens your analytical toolkit.", "From continuous graphs to discrete solutions, ( 2x + 3y = 100 ) demonstrates how basic math concepts scale into powerful practical tools—essential for students, researchers, and professionals alike.", "---", "### Further Reading & Resources", "- Graphing linear equations: Khan Academy – Linear Equations\n- Diophantine equations explained: Brilliant.org – Integer Solutions to Linear Equations\n- Real-world algebra applications: Purdue OWL – Applications of Linear Equations", "---", "Keywords: ( 2x + 3y = 100 ), algebra, linear equation, Diophantine equation, integer solutions, graphing linear equations, budgeting, resource allocation, real-world applications, continuous vs discrete solutions."]
