Question: If $x + y = 10$ and $x^2 + y^2 = 58$, find $x^3 + y^3$.
["Title: How to Solve $x^3 + y^3$ Given $x + y = 10$ and $x^2 + y^2 = 58$ – Step-by-Step Guide", "If you’re solving equations involving pairs of variables like $x + y = 10$ and $x^2 + y^2 = 58$, one common question that arises is: What is $x^3 + y^3$? This problem is a classic algebra challenge and has a neat mathematical shortcut based on known identities.", "---", "### Understanding the Problem", "You’re given:", "- $x + y = 10$\n- $x^2 + y^2 = 58$", "You’re asked to find:\n$x^3 + y^3$", "Don’t get confused by the squared variable — this problem can be solved efficiently using algebraic identities rather than brute-force solving for $x$ and $y$.", "---", "### Use the Identity for $x^3 + y^3$", "There’s a key identity in algebra:\n$$\nx^3 + y^3 = (x + y)^3 - 3xy(x + y)\n$$", "We know $x + y = 10$, so we can plug that in:\n$$\nx^3 + y^3 = (10)^3 - 3xy(10) = 1000 - 30xy\n$$", "Now, the challenge is to find $xy$ — the product of $x$ and $y$.", "---", "### Find $xy$ Using $x^2 + y^2$", "We know the identity:\n$$\nx^2 + y^2 = (x + y)^2 - 2xy\n$$", "Substitute known values:\n$$\n58 = (10)^2 - 2xy = 100 - 2xy\n$$", "Solve for $xy$:\n$$\n2xy = 100 - 58 = 42 \Rightarrow xy = 21\n$$", "---", "### Plug $xy$ Back into the Formula", "Now plug $xy = 21$ into the earlier expression for $x^3 + y^3$:\n$$\nx^3 + y^3 = 1000 - 30(21) = 1000 - 630 = 370\n$$", "---", "### ✅ Final Answer", "$$\n\boxed{x^3 + y^3 = 370}\n$$", "---", "### Why This Works", "This method avoids solving quadratic equations to find $x$ and $y$ explicitly. Instead, it uses symmetric identities that relate sums and products — a powerful technique in algebra and competitive exams.", "---", "### Tips for Similar Problems", "- Always look for identities involving $x + y$ and $x^2 + y^2$ to find $xy$.\n- Express $x^3 + y^3$ using $(x + y)^3 - 3xy(x + y)$.\n- Master substitution — once $xy$ is known, plug in to simplify.", "---", "### Summary", "When given $x + y$ and $x^2 + y^2$, compute $x^3 + y^3$ using:\n$$\nx^3 + y^3 = (x + y)^3 - 3xy(x + y)\n$$", "And always find $xy$ from $x^2 + y^2 = (x + y)^2 - 2xy$.\nThis approach is efficient, elegant, and widely applicable in equations involving symmetric variables.", "---", "Keywords: $x + y = 10$, $x^2 + y^2 = 58$, $x^3 + y^3$, algebraic identity, solve for $xy$, step-by-step math solution, algebra tip, symmetric equations.", "---", "See this identity commonly used in math competitions, high school algebra, and calculus prep — mastering it improves problem-solving speed and accuracy!"]
