If \( a \cdot (a + b) = 20 \) and \( a + b = 7 \), find the value of \( a \).
["## Finding the Value of ( a ): A Step-by-Step Algebra Solution", "Solving equations involving multiple variables may seem challenging at first, but with a clear strategy, even complex problems become manageable. This article walks you through solving the system:", "[\na \cdot (a + b) = 20 \quad \ ext{and} \quad a + b = 7\n]", "to find the value of ( a ). Whether you’re a student, educator, or just someone curious about algebraic methods, this step-by-step guide will help you understand how to uncover unknowns efficiently.", "---", "### Step 1: Understand the Given Equations", "We are given two equations:", "1. ( a \cdot (a + b) = 20 )\n2. ( a + b = 7 )", "These equations are linked because both involve the sum of ( a ) and ( b ). This connection allows us to substitute and simplify the problem.", "---", "### Step 2: Use Substitution to Eliminate One Variable", "From the second equation, we can express ( b ) in terms of ( a ):", "[\nb = 7 - a\n]", "Substitute this expression into the first equation:", "[\na \cdot (a + (7 - a)) = 20\n]", "Simplify the expression inside the parentheses:", "[\na + 7 - a = 7\n]", "So, the equation reduces to:", "[\na \cdot 7 = 20\n]", "---", "### Step 3: Solve for ( a )", "Divide both sides by 7:", "[\na = \frac{20}{7}\n]", "---", "### Step 4: Verify the Solution (Optional but Recommended)", "Plug ( a = \frac{20}{7} ) back into ( a + b = 7 ) to find ( b ):", "[\n\frac{20}{7} + b = 7 \quad \Rightarrow \quad b = 7 - \frac{20}{7} = \frac{49 - 20}{7} = \frac{29}{7}\n]", "Now check the original equation:", "[\na(a + b) = \frac{20}{7} \cdot \left(\frac{20}{7} + \frac{29}{7}\right) = \frac{20}{7} \cdot \frac{49}{7} = \frac{20 \cdot 49}{49} = 20\n]", "✓ The solution satisfies both equations.", "---", "### Why This Method Works", "By substituting ( b = 7 - a ), we eliminated ( b ) entirely, reducing the system to a single equation in ( a ). This technique is powerful for solving simultaneous equations where one variable can be expressed in terms of the other.", "---", "### Key Takeaways", "- Always start by identifying connections between equations.\n- Substitution simplifies complex systems by replacing one variable.\n- Once you solve for one variable, substitute back to verify your result.", "---", "### Final Answer", "[\n\boxed{a = \frac{20}{7}}\n]", "This value of ( a ) satisfies the given equations and demonstrates how thoughtful algebraic manipulation leads to efficient problem-solving.", "---", "Keywords: solve equations, algebra, find value of ( a ), substitution method, system of equations, ( a(a + b) = 20 ), ( a + b = 7 ), step-by-step solution, mathematical problem-solving, algebra tutorial."]
