From the first equation, $ b = 15 - a $. Substitute into the second:
["SEO-Optimized Article: Solving the First Equation and Substitution in Algebra", "Understanding basic algebraic techniques is essential for solving equations efficiently. One foundational skill involves manipulating equations through substitution—a process crucial for simplifying systems of equations. This article explores the step-by-step method using the system:", "From the first equation:\n$ b = 15 - a $", "We’ll demonstrate how to substitute this expression into a second equation to solve for one variable in terms of another. Whether you're a high school student mastering algebra or a lifelong learner brushing up on core math concepts, mastering substitution unlocks more complex problem-solving abilities.", "---", "### The Starting Point: $ b = 15 - a $", "The equation $ b = 15 - a $ expresses $ b $ in terms of $ a $, indicating a direct linear relationship between the two variables. This form is especially useful because it allows us to substitute $ b $ whenever the variable appears in another equation—key for solving systems.", "This simple expression reveals how $ b $ decreases as $ a $ increases—a foundational understanding for graphing linear functions and analyzing relationships in real-world applications.", "---", "### Substitution: Plugging in to Simplify", "Substitution works by replacing the known expression into a second equation. For example:", "Second equation (example):\n$ 2a + b = 35 $", "Instead of solving both equations simultaneously, substitute $ b = 15 - a $ directly into this equation:", "$$\n2a + b = 35\n\Rightarrow 2a + (15 - a) = 35\n$$", "Now simplify:", "$$\n2a + 15 - a = 35\n\Rightarrow (2a - a) + 15 = 35\n\Rightarrow a + 15 = 35\n$$", "---", "### Solving the Resulting Equation", "Continue simplifying:", "$$\na = 35 - 15\n\Rightarrow a = 20\n$$", "Now that we have $ a $, substitute back to find $ b $:", "$$\nb = 15 - a = 15 - 20 = -5\n$$", "---", "### Final Answer", "The solution to the system is:\n$ a = 20 $, $ b = -5 $", "This example illustrates how substitution simplifies problem-solving by reducing the system to a single variable, making calculation straightforward and less error-prone.", "---", "### Why Substitution Matters in Algebra", "- Saves Time: Eliminates the need for elimination or graphical sketching by reducing variables.\n- Improves Clarity: Step-by-step substitution enhances mental math and understanding.\n- Applies Beyond Classrooms: Used in economics, engineering, and computer science for modeling relationships.", "Mastering this core algebraic technique prepares learners for advanced topics like simultaneous equations, linear programming, and function analysis.", "---", "### Practice Makes Perfect", "Try solving your own equations using substitution with different initial equations:", "- $ c = 7 + 3d $ into $ c + d = 25 $\n- $ p = 12 - 2q $ in $ 3p - q = 10 $", "Each practice reinforces the logic behind variable replacement.", "---", "Key Takeaways:\n- Always start by isolating one variable when possible.\n- Substitute directly to eliminate $ b $ or $ c $ from two-variable systems.\n- Simplify carefully—each step maintains equation balance.\n- Back-substitute to find full solution sets.", "Embrace substitution as a powerful tool to turn complex problems into manageable steps. The first equation $ b = 15 - a $ is just the beginning—each substitution builds confidence for more advanced algebra. Start solving smarter today!", "---", "関連キーワード（SEO Keywords）:\n代数練習, 方程式の解法, 代入法, 方程式の解き方, 線形方程式, 代数スキル, 方程式の応用, 数学の基礎, 代入代入法, 1枚の式 Variables substitution tutorial", "---", "Ready to strengthen your algebra skills? Practice substitution daily—your equation-solving confidence will grow step by step."]
