\( 120x + 180(30 - x) = 7800 \)

["# Solving the Linear Equation: 120x + 180(30 - x) = 7800", "Solving linear equations is a foundational skill in algebra that applies widely in real-world scenarios such as budgeting, engineering, and economics. One common equation encountered is:", "[\n120x + 180(30 - x) = 7800\n]", "In this article, we’ll break down the step-by-step solution to this equation, explain the logic behind each step, and highlight how mastering such problems strengthens analytical thinking and problem-solving skills.", "---", "## Understanding the Equation", "The equation\n[\n120x + 180(30 - x) = 7800\n]\nrepresents a real-life situation where variables, often denoted by ( x ), denote unknown quantities—in many cases, price, quantity, or time. Here, ( x ) typically represents an unknown value, and the goal is to isolate ( x ) and determine its value.", "Expanding and simplifying the expression will lead us to a clear solution.", "---", "## Step-by-Step Solution", "### Step 1: Expand the Parentheses", "Start by distributing the 180 into the parentheses:", "[\n120x + 180(30 - x) = 120x + 5400 - 180x\n]", "### Step 2: Combine Like Terms", "Combine the ( x )-terms:", "[\n120x - 180x + 5400 = -60x + 5400\n]", "So the equation now becomes:", "[\n-60x + 5400 = 7800\n]", "### Step 3: Isolate the Term with ( x )", "Subtract 5400 from both sides:", "[\n-60x = 7800 - 5400\n]\n[\n-60x = 2400\n]", "### Step 4: Solve for ( x )", "Divide both sides by -60:", "[\nx = \frac{2400}{-60} = -40\n]", "Wait! This gives ( x = -40 ), which may raise an immediate red flag—can a quantity be negative?", "Let’s double-check translating real-world meaning. In many contexts (such as pricing or time), ( x ) must be non-negative. But algebra itself allows negative solutions; interpretation depends on context.", "---", "## Interpretation and Contextual Consideration", "The negative result ( x = -40 ) suggests the original setup may model a scenario with constraints: For instance, ( x ) might represent a discount or a reduction from an ideal value. If negative, we assess whether constraints are violated.", "However, mathematically, the solution process is correct. The equation simplifies directly to:", "[\nx = -40\n]", "So, formally:", "Answer:\n[\nx = -40\n]", "---", "## Verifying the Solution", "Plug ( x = -40 ) back into the original equation:", "Left-hand side:\n[\n120(-40) + 180(30 - (-40)) = -4800 + 180(70) = -4800 + 12600 = 7800\n]", "Which matches the right-hand side, confirming the solution is correct.", "---", "## Why This Equation Matters", "Learning to solve equations like\n[\n120x + 180(30 - x) = 7800\n]\nprepares learners to model and solve real-world problems, such as:", "- Break-even analysis in business\n- Mixture problems involving concentrations\n- Scheduling and resource allocation", "Mastering step-by-step isolation of variables builds confidence and analytical rigor.", "---", "## Summary", "- Expand: ( 120x + 180(30 - x) = 120x + 5400 - 180x )\n- Combine terms: ( -60x + 5400 = 7800 )\n- Solve: ( -60x = 2400 \Rightarrow x = -40 )\n- Check: Plug back to verify correctness", "Though a negative ( x ) may be unusual, the derivation is sound. Context guides whether such a solution makes practical sense.", "---", "## Want to Practice More?", "Try similar problems involving real-world word equations—such as combining fixed and variable terms—to strengthen your algebra skills. platform", "---", "### Keywords:\n120x + 180(30 - x) = 7800, linear equations, algebra tutorial, solving equations step-by-step, algebra problem-solving, real-world equations, math practice, algebra for beginners", "---", "Further Reading:\n- How to solve word problems involving linear expressions\n- Applications of linear equations in business and science\n- Common mistakes when solving equations like this", "---", "Understanding and solving equations like this equips you with tools to tackle complex challenges across STEM fields—making algebra not just arithmetic, but a gateway to logical thinking and real impact."]