3x + 2y = 24 \quad \text{(1)}\\

["# Solving the Linear Equation: 3x + 2y = 24 (1) — A Complete Guide", "Understanding and solving linear equations is a fundamental skill in algebra, widely used in math, science, economics, and everyday practical applications. One of the most commonly encountered equations is 3x + 2y = 24 (Equation 1). This article provides a detailed step-by-step guide to solving this equation, explores its applications, and introduces key methods and concepts you need to master.", "---", "## Understanding the Equation: 3x + 2y = 24", "The equation 3x + 2y = 24 is a linear Diophantine equation with two variables, x and y. It describes a straight line in a coordinate plane and represents an infinite set of solutions (x, y) that satisfy the relationship. Such equations are essential in modeling real-world scenarios such as budget constraints, mixture problems, and optimization problems across many disciplines.", "---", "## Step-by-Step Guide to Solving 3x + 2y = 24", "### Step 1: Express One Variable in Terms of the Other", "To solve for one variable, isolate it using algebraic manipulation. For example, solving for y:", "$$\n3x + 2y = 24 \\n\Rightarrow 2y = 24 - 3x \\n\Rightarrow y = \frac{24 - 3x}{2}\n$$", "This expresses y directly in terms of x. Similarly, solving for x:", "$$\n3x = 24 - 2y \\n\Rightarrow x = \frac{24 - 2y}{3}\n$$", "---", "### Step 2: Identify Whole Number Solutions (Integer Solutions)", "Often, in word problems, we seek integer solutions. To find valid x, y pairs, substitute integer values of x and check whether y becomes an integer.", "Since y = (24 - 3x)/2, 24 - 3x must be divisible by 2. That means 3x must be even ⇒ x must be even (since 3 is odd, odd × even = even).", "### Try Even Integer Values for x:", "| x | 3x | 24 - 3x | y = (24 - 3x)/2 | Solution (x, y) |\n|----|-----|---------|------------------|---------------------|\n| 0 | 0 | 24 | 12 | (0, 12) |\n| 2 | 6 | 18 | 9 | (2, 9) |\n| 4 | 12 | 12 | 6 | (4, 6) |\n| 6 | 18 | 6 | 3 | (6, 3) |\n| 8 | 24 | 0 | 0 | (8, 0) |\n| 10 | 30 | -6 | –3 | Invalid y (negative) |\n| — | — | — | — | — |", "Thus, integer solutions occur when x = 0, 2, 4, 6, 8, yielding y = 12, 9, 6, 3, 0 respectively.", "---", "### Step 3: Graphing the Equation", "Plotting 3x + 2y = 24 confirms a straight line through the integer points listed. The y-intercept is (0, 12), and the x-intercept is (8, 0). The slope Δy/Δx = –3/2, showing how y decreases as x increases by 2 units.", "---", "## Applications of 3x + 2y = 24", "- Budgeting & Finance: If 'x' represents units of product A costing $3 and 'y' units of product B costing $2, the equation models a $24 budget constraint.\n- Mixing Solutions: Used in chemistry to describe mixtures, e.g., combining two solutions with cost properties.\n- Resource Allocation: Modeling scenarios where two limited resources (x and y) contribute to a total capacity or cost.", "---", "## Advanced Approaches: General Solution Set", "The general solution to 3x + 2y = 24 includes all integer points derived from:", "$$\ny = 12 - \frac{3}{2}x\n$$", "To generate all real solutions, x can take any real value such that y is real, but practical contexts often require x, y ∈ ℤ⁺ (positive integers). From the table above, valid pairs are:", "[\n(x, y) = (2k, 24 - 6k) \quad \ ext{for integer } k \ ext{ such that } x > 0, y \geq 0\n]", "Making ( k = 0, 1, 2, 3, 4 ) gives the five solutions listed earlier.", "---", "## Why Mastering This Equation Matters", "- Builds foundational algebraic reasoning.\n- Prepares learners for systems of equations and linear programming.\n- Applies directly to economics, engineering, computer science (e.g., constraint programming).\n- Demonstrates how abstract math models real-life decision-making.", "---", "## Conclusion", "Solving 3x + 2y = 24 elects not just finding a numerical answer—but understanding relationships between variables, iterating through possible values, and applying algebra to model real-world situations. Whether you're solving algebra homework, optimizing a plan, or analyzing data, mastering such equations is essential.", "For further practice, explore alternative forms (slope-intercept: ( y = 12 - \frac{3}{2}x )), examine systems of equations, or dive into graphing technologies like Desmos to visualize solution sets.", "---", "### Keywords for SEO:\n- 3x + 2y = 24 solutions\n- How to solve 3x + 2y = 24\n- Integer solutions to 3x + 2y = 24\n- Algebra linear equation tutorial\n- Real-world applications of 3x + 2y = 24\n- Graphing and solving linear equations", "---", "Keywords Summary:\n3x + 2y = 24, linear equation, algebra, solve for y, integer solutions, graphing, real-world applications, systems of equations, intercept form."]