We are given \(m + n = 7\) and \(m^2 + n^2 = 29\).

["Unlocking Solutions: Solving the System (m + n = 7) and (m^2 + n^2 = 29)", "Understanding how to solve systems of equations is a foundational skill in algebra and beyond. In this article, we explore how to determine the values of (m) and (n) given the constraints:\n- ( m + n = 7 )\n- ( m^2 + n^2 = 29 )", "These simple-looking equations open a pathway to deeper mathematical reasoning involving substitution, algebraic identities, and even real-world interpretation. Let’s break down the process step by step.", "### Step 1: Use the Sum Constraint to Express One Variable\nThe equation ( m + n = 7 ) provides a clear entry point. We can express ( n ) in terms of ( m ):\n[\nn = 7 - m\n]", "Substituting this into the second equation ( m^2 + n^2 = 29 ), we get:\n[\nm^2 + (7 - m)^2 = 29\n]", "### Step 2: Expand and Simplify\nNow expand ( (7 - m)^2 ):\n[\n(7 - m)^2 = 49 - 14m + m^2\n]\nSubstituting back:\n[\nm^2 + 49 - 14m + m^2 = 29\n]\nCombine like terms:\n[\n2m^2 - 14m + 49 = 29\n]\nSubtract 29 from both sides:\n[\n2m^2 - 14m + 20 = 0\n]\nDivide the entire equation by 2 to simplify:\n[\nm^2 - 7m + 10 = 0\n]", "### Step 3: Solve the Quadratic Equation\nWe now solve the quadratic ( m^2 - 7m + 10 = 0 ) using factoring.\nWe look for two numbers that multiply to (10) and add to (-7). These are (-5) and (-2):\n[\n(m - 5)(m - 2) = 0\n]", "Setting each factor equal to zero gives the solutions:\n[\nm = 5 \quad \ ext{or} \quad m = 2\n]", "### Step 4: Find Corresponding (n) Values\nUsing ( m + n = 7 ), find (n) for each (m):\n- If ( m = 5 ): ( n = 7 - 5 = 2 )\n- If ( m = 2 ): ( n = 7 - 2 = 5 )", "Thus, the solution set is the ordered pair ( (5, 2) ) and ( (2, 5) ).", "### Step 5: Verify Using Algebraic Impulse\nBeyond arithmetic substitution, note that the identity:\n[\n(m + n)^2 = m^2 + n^2 + 2mn\n]\nlets us quickly check:\n[\n7^2 = 29 + 2mn \Rightarrow 49 = 29 + 2mn \Rightarrow 2mn = 20 \Rightarrow mn = 10\n]\nThe product ( mn = 10 ). For ( (5,2) ) and ( (2,5) ), both pairs satisfy ( m+n = 7 ) and ( mn = 10 ), confirming our solutions are consistent.", "### Why This Matters: Applications and Insight\nSolving equations of the form ( m + n = s ) and ( m^2 + n^2 = t ) appears in physics (e.g., energy conservation), economics (budget models), and computer science (constraint satisfaction). This problem illustrates how a system of two equations delivers precise numerical results and reinforces algebraic substitution as a powerful technique.", "### Summary\nGiven:\n[\nm + n = 7\n]\n[\nm^2 + n^2 = 29\n]\nThe solutions are:\n[\n(m, n) = (5, 2) \quad \ ext{or} \quad (2, 5)\n]\nUse substitution and identities to solve such systems efficiently — a skill useful across STEM disciplines.", "---\nFor more algebraic problem-solving tips and step-by-step explanations, explore our dedicated algebra guides and video tutorials!"]