\frac{4d + 3d}{240} = 7 \Rightarrow \frac{7d}{240} = 7
["Solving the Equation: \frac{4d + 3d}{240} = 7 → \frac{7d}{240} = 7", "In algebra, solving linear equations is a fundamental skill that helps quantify relationships and find unknown values. One common type of equation students and professionals encounter involves simplifying expressions before solving. Consider the equation:", "[\n\frac{4d + 3d}{240} = 7\n]", "At first glance, this equation appears straightforward, but mastering it involves combining like terms and isolating variables effectively. Let’s walk through the step-by-step solution and explain why transforming it to (\frac{7d}{240} = 7) is a key simplification.", "---", "### Step 1: Combine Like Terms in the Numerator", "The numerator (4d + 3d) consists of two terms with the same variable (d). Combine them:", "[\n4d + 3d = 7d\n]", "So the equation becomes:", "[\n\frac{7d}{240} = 7\n]", "This simplification reduces complexity, making it easier to isolate (d).", "---", "### Step 2: Understanding the Fraction Equality", "We now have (\frac{7d}{240} = 7), meaning 7d divided by 240 equals 7. To solve for (d), eliminate the denominator by multiplying both sides of the equation by 240.", "[\n7d = 7 \ imes 240\n]", "Calculate the right-hand side:", "[\n7 \ imes 240 = 1680\n]", "So:", "[\n7d = 1680\n]", "---", "### Step 3: Solve for (d)", "Now, divide both sides by 7 to isolate (d):", "[\nd = \frac{1680}{7} = 240\n]", "---", "### Step 4: Verify the Solution", "Substitute (d = 240) into the original equation to confirm:", "[\n\frac{4(240) + 3(240)}{240} = \frac{960 + 720}{240} = \frac{1680}{240} = 7\n]", "The left-hand side equals the right-hand side, confirming the solution is correct.", "---", "### Why This Simplification Matters", "Rewriting (\frac{4d + 3d}{240} = 7) as (\frac{7d}{240} = 7) demonstrates a core principle in algebra: simplifying expressions before solving. This reduces messy denominators early and accelerates solving steps. Mastering such transformations helps students and learners build fluency with linear equations and prepare for more complex algebra.", "---", "### Summary", "- Combine like terms in the numerator to simplify.\n- Eliminate denominators carefully to isolate the variable.\n- Always verify your solution by substituting back into the original equation.\n- Simplifying fractions early leads to more efficient problem-solving.", "Key takeaway: The equation (\frac{4d + 3d}{240} = 7) simplifies neatly to (\frac{7d}{240} = 7), a key step in solving for (d = 240).", "---", "For more algebraic techniques and step-by-step guides, explore our full library on solving linear equations, fraction equations, and equation simplification strategies. Accurate and efficient solving begins with clarity—start by simplifying the expression before executing calculations."]
