The difference between the squares of two numbers is 45, and their sum is 13. What are the numbers?

["The Difference Between the Squares of Two Numbers Is 45, and Their Sum Is 13 — What Are the Numbers?", "If you’ve ever come across a puzzle involving squares of numbers, the story of two numbers whose squares differ by 45 and sum to 13 reveals an elegant mathematical relationship. Let’s break down how to solve this intriguing problem step by step.", "---", "### Understanding the Problem", "We are given two key conditions:", "1. The difference between the squares of two numbers ( x ) and ( y ) is 45:\n [ x^2 - y^2 = 45 ]", "2. The sum of the numbers is 13:\n [ x + y = 13 ]", "We want to find the actual values of ( x ) and ( y ).", "---", "### Why This Approach Works", "The expression ( x^2 - y^2 ) is a classic algebraic identity — it factors as the product of a sum and a difference:\n[ x^2 - y^2 = (x + y)(x - y) ]", "This identity is the key to solving the problem efficiently, combining both given conditions.", "---", "### Step 1: Use the Sum to Express One Variable", "From the sum:\n[ x + y = 13 ]\nWe can express ( x ) as:\n[ x = 13 - y ]", "Now substitute this expression into the difference-of-squares equation.", "---", "### Step 2: Substitute into ( x^2 - y^2 = 45 )", "Using the identity:\n[ (x + y)(x - y) = 45 ]", "We already know ( x + y = 13 ), so:\n[ 13(x - y) = 45 ]\nSolve for ( x - y ):\n[ x - y = \frac{45}{13} ]", "But now we have two equations:", "- ( x + y = 13 )\n- ( x - y = \frac{45}{13} )", "These can be added and subtracted to find ( x ) and ( y ).", "---", "### Step 3: Solve the System of Equations", "Add the two equations:\n[ (x + y) + (x - y) = 13 + \frac{45}{13} ]\n[ 2x = \frac{169}{13} + \frac{45}{13} = \frac{214}{13} ]\n[ x = \frac{107}{13} ]", "Now subtract the equations:\n[ (x + y) - (x - y) = 13 - \frac{45}{13} ]\n[ 2y = \frac{169 - 45}{13} = \frac{124}{13} ]\n[ y = \frac{62}{13} ]", "At first glance, these are fractional values. However, let’s re-evaluate using integer possibilities, since such puzzles often involve whole numbers.", "---", "### Step 4: Try Integer Solutions", "Let’s reconsider the original clues with a focus on integer values — common in classic number puzzles.", "Let ( x ) and ( y ) be integers such that:\n[ x + y = 13 ]\n[ x^2 - y^2 = 45 ]", "Recall:\n[ x^2 - y^2 = (x + y)(x - y) = 45 ]\nSubstitute ( x + y = 13 ):\n[ 13(x - y) = 45 ]\n[ x - y = \frac{45}{13} ] — not an integer.", "But wait — 45 and 13 are coprime, so unless we made a mistake, the numbers may not be integers.", "Wait — let's double-check:\nIs there a pair of integers satisfying both?", "Let’s test small integer pairs adding to 13:", "- ( (10, 3): 10^2 - 3^2 = 100 - 9 = 91 <br/>\ne 45 )\n- ( (9, 4): 81 - 16 = 65 )\n- ( (8, 5): 64 - 25 = 39 )\n- ( (7, 6): 49 - 36 = 13 )\n- ( (11, 2): 121 - 4 = 117 )\n- ( (12, 1): 144 - 1 = 143 )", "None equal 45.", "Wait — this suggests no integer solution exists with sum 13 and difference of squares 45.", "But let’s go back — maybe we misapplied logic?", "Wait — actually, there is a unique real solution, even if not integer.", "Let’s return to the algebra:", "From:\n[ x + y = 13 ]\n[ x^2 - y^2 = 45 ]\nUse identity:\n[ x^2 - y^2 = (x + y)(x - y) ]\n[ 45 = 13(x - y) ]\n[ x - y = \frac{45}{13} ]", "Now solve:\n[ x + y = 13 ]\n[ x - y = \frac{45}{13} ]", "Add:\n[ 2x = 13 + \frac{45}{13} = \frac{169 + 45}{13} = \frac{214}{13} ]\n[ x = \frac{107}{13} ]\n[ y = 13 - \frac{107}{13} = \frac{169 - 107}{13} = \frac{62}{13} ]", "---", "### Final Answer: The Numbers Are ( \frac{107}{13} ) and ( \frac{62}{13} )", "These are the only real numbers satisfying both conditions.", "---", "### Why This Matters", "This problem illustrates a powerful application of algebraic identities. Even when numbers aren’t integers, algebra allows us to precisely determine solutions — a key skill in solving real-world and theoretical math puzzles.", "---", "### Summary", "| Condition | Equation | Result |\n|------------------------|----------------------------------|----------------------------|\n| Sum of numbers | ( x + y = 13 ) | Given |\n| Difference of squares | ( x^2 - y^2 = 45 ) | Use identity: ( (x+y)(x-y) = 45 ) |\n| Substitute sum: ( 13(x - y) = 45 ) | ( x - y = \frac{45}{13} ) | |\n| Solve system: | ( x + y = 13 ), ( x - y = \frac{45}{13} ) | ( x = \frac{107}{13} ), ( y = \frac{62}{13} ) |", "---", "### Conclusion", "The difference of the squares being 45 and their sum being 13 leads uniquely to the fractions ( \frac{107}{13} ) and ( \frac{62}{13} ). This example proves how algebraic methods uncover elegant solutions, even in puzzles that first seem exclusive to whole numbers.", "If you're tackling similar problems, remember: factor identities and system solving are your allies — and sometimes, the truth is more elegant than the obvious!", "---", "Keywords: difference of squares problem, two numbers difference of squares equals 45, sum is 13, algebraic solution, integer and fractional solutions, ( x^2 - y^2 = (x+y)(x-y) ), real number solutions, puzzle math.\nMeta Title: The difference between the squares of two numbers is 45, and their sum is 13 — what are the numbers?\nMeta Description: Solve the number puzzle: squares differ by 45, sum is 13 — find the exact values of the numbers using algebra.\nHeader Tags: #DifferenceOfSquares #AlgebraProblem #NumberPuzzleSolution"]