\[ 400a + 20b = 48 \]

["Understanding the Linear Equation: 400a + 20b = 48", "Solving linear equations is fundamental in algebra, and understanding equations like ( 400a + 20b = 48 ) opens doors to numerous real-world applications—from budgeting and business modeling to science and engineering. This comprehensive guide explores the equation ( 400a + 20b = 48 ), revealing how to interpret, solve, and apply it in practical scenarios.", "---", "### What is the Equation ( 400a + 20b = 48 )?", "The equation ( 400a + 20b = 48 ) is a linear Diophantine equation involving two variables ( a ) and ( b ). It expresses a linear relationship between ( a ) and ( b ), where ( a ) and ( b ) can be any real or integer numbers satisfying the equation.", "This type of equation frequently appears in:", "- Financial planning (e.g., budget allocations with fixed costs)\n- Resource allocation problems\n- Physics problems involving linear relationships\n- Computer algorithms and discrete mathematics", "---", "### Step-by-Step Solution", "Step 1: Simplify the equation\nBegin by simplifying the coefficients. Notice that all terms are divisible by 4:", "[\n400a + 20b = 48 \quad \Rightarrow \quad 100a + 5b = 12\n]", "This simplified form is easier to work with.", "---", "Step 2: Express in terms of one variable", "We can solve for one variable in terms of the other. Solving for ( b ):", "[\n5b = 12 - 100a\n]\n[\nb = \frac{12 - 100a}{5} = \frac{12}{5} - 20a\n]", "---", "Step 3: Account for integer solutions (if required)", "A Diophantine equation seeks integer solutions. For ( b = \frac{12}{5} - 20a ), the expression is rational but not necessarily integer unless ( a ) is chosen so that ( 12 - 100a ) is divisible by 5. Since 100a is always divisible by 5, and 12 mod 5 = 2, we require:", "[\n12 - 100a \equiv 0 \pmod{5} \Rightarrow 2 \equiv 0 \pmod{5}\n]", "This fails, meaning the equation has no integer solutions. However, if ( a ) and ( b ) are real numbers, solutions exist for any real ( a ), with ( b ) determined accordingly.", "---", "Step 4: Graph the line (optional visualization)", "Plotting ( 100a + 5b = 12 ), this represents a straight line in the ( ab )-plane with:", "- ( a )-intercept at ( a = 0.12 ) (when ( b = 0 ))\n- ( b )-intercept at ( b = 2.4 ) (when ( a = 0 ))", "---", "### Applications of ( 400a + 20b = 48 )", "1. Budgeting and Finance:\n Suppose ( a ) represents the number of units of one product costing $400 each, and ( b ) the number of units of another costing $20. The equation models a total cost of $48 — useful for analyzing minimal purchasing levels.", "2. Resource Allocation:\n In scheduling or logistics, variables can represent quantities allocated to different tasks. This equation helps identify combinations meeting a fixed constraint.", "3. Math and Science Problems:\n When modeling linear trends such as speed, force, or chemical concentrations, dual variables like ( a ) and ( b ) may appear under linear conditions.", "---", "### Tips to Solve Such Equations:", "- Always simplify equations by factoring out common terms.\n- Recognize when solutions must be integers and test conditions on modulus.\n- Use substitution or graphing for clarity.\n- Apply real-world interpretation to anchor abstract solutions.", "---", "### Conclusion", "Equation ( 400a + 20b = 48 ) is a simple yet powerful linear relationship. Whether interpreted over real numbers or restricted to integers, understanding how to manipulate and solve it enhances problem-solving skills across disciplines. From allocating budgets to modeling physical systems, tools to handle this equation are essential for both students and professionals.", "---", "Related Keywords for SEO:\nlinear equation solver, 400a + 20b solution, Diophantine equation real solutions, linear algebra applications, real-world math problems, simplifying 400a + 20b, Diophantine equations with real coefficients.", "---", "Need further help with linear equations? Explore our guides on solving systems of equations, Diophantine equations, and applications of algebra in real life!"]