Substitute to get \( 210 = \frac{7}{2}(15 + l) \).

["Substitute to Solve: Solving the Equation ( 210 = \frac{7}{2}(15 + l) )", "Solving linear equations can feel challenging, but using smart substitutions makes the process simpler and more intuitive. In this article, we’ll explore an effective substitute method to solve the equation:\n[ 210 = \frac{7}{2}(15 + l) ]", "---", "### Step 1: Understand the Equation", "Start with the equation:\n[ 210 = \frac{7}{2}(15 + l) ]", "Our goal is to isolate the variable ( l ). The key is to eliminate the fraction and simplify the equation.", "---", "### Step 2: Substitute to Eliminate the Fraction", "Instead of multiplying both sides by ( \frac{2}{7} ), which can be error-prone or cumbersome, we make a substitution to simplify notation and steps.", "Let:\n[ l = x - 15 ]\nThis substitution shifts the expression ( (15 + l) ) into a simpler form.", "Why this substitution?\nSubstituting ( l = x - 15 ) turns ( 15 + l ) into ( x ), reducing the equation to a cleaner linear form.", "---", "### Step 3: Apply the Substitution", "Substitute ( l = x - 15 ) into the original equation:\n[ 210 = \frac{7}{2}(15 + (x - 15)) ]", "Simplify inside the parentheses:\n[ 15 + (x - 15) = x ]", "Now the equation becomes:\n[ 210 = \frac{7}{2}x ]", "---", "### Step 4: Solve the Simplified Equation", "Multiply both sides by ( \frac{2}{7} ) to isolate ( x ):\n[ x = 210 \cdot \frac{2}{7} ]", "Calculate:\n[ x = \frac{420}{7} = 60 ]", "---", "### Step 5: Back-Substitute to Find ( l )", "Recall our substitution:\n[ l = x - 15 ]\nSubstitute ( x = 60 ):\n[ l = 60 - 15 = 45 ]", "---", "### Final Answer:", "[ \boxed{l = 45} ]", "---", "### Bonus Tips for Mastering Substitutions in Equations", "- Substitutions simplify complex expressions into manageable forms.\n- Always verify your solution by plugging ( l = 45 ) back into the original equation:\n [ 210 = \frac{7}{2}(15 + 45) = \frac{7}{2}(60) = 210 ]\n ✔️ Equation holds true!\n- Practice with similar structures: equations involving fractions, parentheses, and coins, distances, or mixtures.", "---", "### Summary", "By using a strategic substitution ( l = x - 15 ), we transformed the equation ( 210 = \frac{7}{2}(15 + l) ) into a straightforward linear equation. This method not only solves the problem cleanly but also reinforces algebraic reasoning essential for more complex equations.", "Master substitution techniques — your key to faster, more confident problem-solving!", "---", "Keywords: substitute to solve equation, solve 210 = 7/2(15 + l), algebraic substitution, linear equation steps, step-by-step equation solving, algebra techniques, solving for l with substitution, simplify equations by substitution."]