Distance: \(30t + 5(4 - t) = 120\).
["# Solving the Linear Equation: ( 30t + 5(4 - t) = 120 )", "Understanding how to solve linear equations is a fundamental skill in algebra, essential for students, educators, and anyone working with mathematical modeling. One common type of problem is expressing real-world scenarios using equations, such as:", "[ 30t + 5(4 - t) = 120 ]", "In this article, we’ll walk through the step-by-step process to solve this equation, explain how to interpret its meaning, and offer practical tips for mastering similar problems.", "## What is the Equation?", "The equation\n[ 30t + 5(4 - t) = 120 ]\nmodels a scenario where ( t ) represents a variable quantity—perhaps time, distance, or a measurable parameter. The expression combines linear terms and parentheses, requiring careful simplification.", "---", "## Step-by-Step Solution", "### Step 1: Expand the Parentheses", "Start by distributing the 5 across the terms inside the parentheses:", "[\n30t + 5(4 - t) = 30t + 20 - 5t\n]", "### Step 2: Combine Like Terms", "Combine the ( t )-terms:", "[\n(30t - 5t) + 20 = 25t + 20\n]", "So the equation becomes:", "[\n25t + 20 = 120\n]", "### Step 3: Isolate the Variable Term", "Subtract 20 from both sides:", "[\n25t = 120 - 20 = 100\n]", "### Step 4: Solve for ( t )", "Divide both sides by 25:", "[\nt = \frac{100}{25} = 4\n]", "---", "## Interpretation and Application", "With ( t = 4 ), we’ve solved for the unknown variable. But what does this mean in context?", "Suppose ( t ) represents time in hours, and the equation models a total cost or distance. For example:", "- You earn $30 per hour and receive a $5 bonus for every 4-hour block over a 4-hour period.\n- The expression accounts for hourly pay plus a structured bonus tied to 4-hour intervals.", "At ( t = 4 ), your total income or accumulated value satisfies the equation.", "---", "## Tips to Solve Similar Linear Equations", "1. Distribute first: Always expand parentheses before combining like terms.\n2. Use order of operations: Parentheses first, then exponents, multiplication/division, then addition/subtraction.\n3. Isolate the variable: Perform inverse operations on both sides to keep the equation balanced.\n4. Check your solution: Substitute ( t = 4 ) back into the original equation to verify correctness.", "---", "## Why This Equation Matters", "Linear equations like ( 30t + 5(4 - t) = 120 ) are widely used in science, economics, and everyday decision-making. Mastering them strengthens your ability to model:", "- Budgeting with fixed and variable costs\n- Physics problems involving rate and time\n- Business projections and break-even analysis", "Understanding how to manipulate and solve such equations empowers you to tackle more advanced mathematical challenges.", "---", "## Conclusion", "Solving ( 30t + 5(4 - t) = 120 ) involves expanding, simplifying, and isolating the variable ( t )—skills that form the backbone of algebra. With practice, algebraic expressions become intuitive tools for analyzing real-world situations. If you’re learning to solve equations, this kind of problem is a stepping stone toward mastery of mathematical reasoning.", "---", "Keywords: linear equation, solve (30t + 5(4 - t) = 120), algebra tutorial, equation solving, distance learning, math practice, algebra examples, solve linear equations, step-by-step math, algebra equation solver, linear expressions.", "---", "Struggling with this or similar equations? Use this guide to build confidence and accuracy—whether for homework, exams, or real-life calculations."]
