We have \( xy = 56 \) and \( x + y = 15 \).
["How to Solve ( xy = 56 ) and ( x + y = 15 ): Step-by-Step Explanation", "When given a system of equations like ( xy = 56 ) and ( x + y = 15 ), many people wonder how to find the values of ( x ) and ( y ). This classic problem appears frequently in algebra and can be efficiently solved using substitution or quadratic equation methods. In this SEO-optimized guide, we’ll explore how to solve these equations, uncovering the values of ( x ) and ( y ), and explaining how to effectively rank this topic in search results for students, educators, and problem solvers.", "### Understanding the Problem", "We are given two equations:", "1. ( x + y = 15 )\n2. ( xy = 56 )", "These are the sum and product of two variables—common in quadratic problems. Recognizing this pattern allows us to solve the system elegantly.", "---", "## Solving Using Substitution or Vieta’s Formula", "Since ( x ) and ( y ) are roots of a quadratic equation, we use the identity:\nIf ( x ) and ( y ) satisfy ( x + y = S ) and ( xy = P ), they are roots of\n[\nt^2 - St + P = 0\n]", "### Step 1: Set up the quadratic equation\nUsing ( S = 15 ) and ( P = 56 ), the equation becomes:\n[\nt^2 - 15t + 56 = 0\n]", "### Step 2: Solve the quadratic equation\nWe factor the quadratic:\nFind two numbers that multiply to ( 56 ) and add to ( 15 ).\nChecking factors of 56:\n( 7 ) and ( 8 ) work since:\n( 7 \ imes 8 = 56 ) and ( 7 + 8 = 15 )", "Thus,\n[\nt^2 - 15t + 56 = (t - 7)(t - 8) = 0\n]", "### Step 3: Find the solutions\nSet each factor to zero:\n( t = 7 ) or ( t = 8 )", "So, ( x = 7 ), ( y = 8 ) or vice versa.", "---", "## Verification", "Check in the original equations:\n- ( x + y = 7 + 8 = 15 ) ✓\n- ( xy = 7 \ imes 8 = 56 ) ✓", "The solution satisfies both conditions.", "---", "## Alternative: Direct Substitution", "Alternatively, express ( y = 15 - x ) from the first equation and substitute into the second:\n[\nx(15 - x) = 56\n\Rightarrow 15x - x^2 = 56\n\Rightarrow x^2 - 15x + 56 = 0\n]", "This leads to the same quadratic and the same solutions.", "---", "## Why This Problem Matters (SEO Relevance)", "Solving simultaneous equations involving sum and product is foundational in algebra, applicable to real-life scenarios like finance, physics, and computer science. Search queries such as “solve for x and y when x + y = 15 and xy = 56” reflect student learning needs. This method is frequently sought by secondary school and university students, educators preparing lessons, and self-learners using search engines.", "Keywords & Phrases:\n- Solve ( xy = 56 ) and ( x + y = 15 )\n- How to find x and y given sum and product\n- Quadratic equations sum and product roots\n- Algebra problem solving tutorial\n- Solve system of equations step-by-step", "Meta Description for SEO:\nLearn how to solve ( xy = 56 ) and ( x + y = 15 ) using substitution and quadratic methods. Step-by-step guide for students and math enthusiasts with verified solutions.", "---", "## Summary", "Given ( x + y = 15 ) and ( xy = 56 ):\n- The values ( x = 7 ), ( y = 8 ) (or vice versa) satisfy both equations.\n- This system is solved by forming a quadratic equation or substitution.\n- Understanding this pattern strengthens algebraic skills widely used in academic and real-world contexts.", "---", "Need more algebra help? Explore related topics like quadratic roots, Vieta’s formulas, and systems of equations to master essential math concepts."]
