messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Work through and solve this equation. Prove that $(\\mathbb{Z}_14, +)$ is a group under addition modulo 14."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_14=\\{0,1,\\dots,13\\}$ with operation $+$ modulo $14$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7500
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. 40 - 21x + 50x^2 = 170"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 40 - 21x + 50x^2 = 170 $\n\nMove all terms to the left (standard form):\n$ 50x^2 + -21x + -130 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7501
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Solve for $x$ (in radians): cos(x) = -\\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -\\frac{1}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arc... | {
"type": "trigonometry",
"seed": 1337,
"index": 7502
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. Evaluate: ((-121 + -84/14) * 4) + 8^3 - log_2(32) + (21 mod 3) + floor(-193/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-121 + -84/14) * 4) + 8^3 - log_2(32) + (21 mod 3) + floor(-193/2) $\n\nCompute ea... | {
"type": "arithmetic",
"seed": 1337,
"index": 7503
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Evaluate: ((54 + 912/24) * 19) + 8^2 - log_10(10000) + (300 mod 15) + round(69/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((54 + 912/24) * 19) + 8^2 - log_10(10000) + (300 mod 15) + rou... | {
"type": "arithmetic",
"seed": 1337,
"index": 7504
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. 44 - 107 + 107x = -45x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 44 - 107 + 107x = -45x^2 $\n\nMove all terms to the left (standard form):\n$ 45x^2 + 107x + -63 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7505
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. 46x^2 + 171 - 146x = -73"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 46x^2 + 171 - 146x = -73 $\n\nMove all terms to the left (standard form):\n$ 46x^2 + -146x + 244 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7506
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Differentiate with respect to $x$: $f(x) = \\ln(6x + -6)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(6x + -6) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\fra... | {
"type": "calculus",
"seed": 1337,
"index": 7507
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. Compute the indefinite integral: $\\int 6x^3 + -3x^2 + -18x + 11\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 6x^3 + -3x^2 + -18x + 11\\,dx $\n\nUse standard antiderivatives (and in... | {
"type": "calculus",
"seed": 1337,
"index": 7508
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. -823 + 485 = 93x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -823 + 485 = 93x $\n\nMove all terms to the left (standard form):\n$ -93x + -338 = 0 $\n\nIsolate the $x$ term:\n$ -93x = 338 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 7509
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\... | {
"type": "optimization",
"seed": 1337,
"index": 7510
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Solve for $x$ (in radians): tan(x) = 1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = 1 $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(1) = 0.785398 $\n\nThe ge... | {
"type": "trigonometry",
"seed": 1337,
"index": 7511
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. Evaluate: ((-170 + -775/25) * 14) + 10^5 - log_9(59049) + (-333 mod 2) + floor(-21/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-170 + -775/25) * 14) + 10^5 - log_9(59049) + (-333 mod 2) + floor(-... | {
"type": "arithmetic",
"seed": 1337,
"index": 7512
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Evaluate: ((-90 + 374/17) * 19) + 12^3 - log_10(10000) + (80 mod 21) + round(-180/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-90 + 374/17) * 19) + 12^3 - log_10(10000) + (80 mod 21) + ... | {
"type": "arithmetic",
"seed": 1337,
"index": 7513
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. Differentiate with respect to $x$: $f(x) = -8\\sin(1x) + 9\\cos(4x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -8\\sin(1x) + 9\\cos(4x) $\n\nApply the standard differentiation r... | {
"type": "calculus",
"seed": 1337,
"index": 7514
} |
[
{
"role": "user",
"content": "Please solve this math problem. 44x^2 - 174 + 129x = -48"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 44x^2 - 174 + 129x = -48 $\n\nMove all terms to the left (standard form):\n$ 44x^2 + 129x + -126 = 0 $\n\nApply the quadratic formula:\n$ x = \\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7515
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. Differentiate with respect to $x$: $f(x) = -2x^3 + -7x^2 + -9x + 6$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -2x^3 + -7x^2 + -9x + 6 $\n\nApply the standard differentiation rules:\n$ f'(... | {
"type": "calculus",
"seed": 1337,
"index": 7516
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. -376x + 791 = -481"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -376x + 791 = -481 $\n\nMove all terms to the left (standard form):\n$ -376x + 1272 = 0 $\n\nIsolate the $x$ term:\n$ -376x = -1272 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7517
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Solve for $x$ (in radians): cos(x) = 1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = 1 $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(1) = 0 $\n\nThe general s... | {
"type": "trigonometry",
"seed": 1337,
"index": 7518
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, hence:\n... | {
"type": "optimization",
"seed": 1337,
"index": 7519
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, hence... | {
"type": "optimization",
"seed": 1337,
"index": 7520
} |
[
{
"role": "user",
"content": "solve Prove that $f(x,y)=\\frac12(10x^2+9y^2)+4x+5y+-4$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(10x^2+9y^2)+4x+5y+-4 $\n\n$ \\frac{\\partial^2 f}{\\partial x^2} = 10,\\quad \\frac{\\p... | {
"type": "optimization",
"seed": 1337,
"index": 7521
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Prove that $f(x,y)=\\frac12(8x^2+8y^2)+-5x+-2y+-5$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(8x^2+8y^2)+-5x+-2y+-5 $\n\n$ \\frac{\\p... | {
"type": "optimization",
"seed": 1337,
"index": 7522
} |
[
{
"role": "user",
"content": "Please solve this math problem. Solve for $x$ (in radians): cos(x) = 0."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = 0 $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(0) = 1.5708 $\n\nThe general solu... | {
"type": "trigonometry",
"seed": 1337,
"index": 7523
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. 15x^2 + 156 + 83 = -195x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 15x^2 + 156 + 83 = -195x $\n\nMove all terms to the left (standard form):\n$ 15x^2 + 195x + 239 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7524
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Differentiate with respect to $x$: $f(x) = 3\\sin(8x) + -1\\cos(1x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 3\\sin(8x) + -1\\cos(1x) $\n\nApply the standard differentiation rules... | {
"type": "calculus",
"seed": 1337,
"index": 7525
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-1&-4\\\\\\-1&0\\end{pmatrix},\\quad b=\\begin{pmatrix}15\\\\\\-9\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-1&-4\\\\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7526
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. Solve for $x$ (in radians): cos(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal valu... | {
"type": "trigonometry",
"seed": 1337,
"index": 7527
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Compute the indefinite integral: $\\int (-2x + -10)e^{4x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-2x + -10)e^{4x}\\,dx $\n\nUse standard antiderivatives (and integration by part... | {
"type": "calculus",
"seed": 1337,
"index": 7528
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Prove that $f(x)=1x^2+-5x+-4$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=1x^2+-5x+-4$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$... | {
"type": "optimization",
"seed": 1337,
"index": 7529
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\... | {
"type": "optimization",
"seed": 1337,
"index": 7530
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Compute the indefinite integral: $\\int (-4x + 8)\\sin(4x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-4x + 8)\\sin(4x)\\,dx $\n\nUse standard antiderivatives (and in... | {
"type": "calculus",
"seed": 1337,
"index": 7531
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Differentiate with respect to $x$: $f(x) = \\ln(5x + 7)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(5x + 7) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{5}{... | {
"type": "calculus",
"seed": 1337,
"index": 7532
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. 850 - 507 = 195x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 850 - 507 = 195x $\n\nMove all terms to the left (standard form):\n$ -195x + 343 = 0 $\n\nIsolate the $x$ term:\n$ -195x = -343 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7533
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Evaluate: ((88 + -966/23) * 3) + 10^4 - log_2(8) + (-33 mod 16) + round(-128/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((88 + -966/23) * 3) + 10^4 - log_2(8) + (-33 mod 16) + round(-128/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 7534
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Compute $(-77+-63) \\bmod 5$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-77+-63) \\bmod 5 $\n\nUse the rule that addition is compatible with reduction mod... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7535
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Evaluate: ((-17 + 720/20) * 18) + 6^4 - log_7(117649) + (368 mod 19) + floor(16/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-17 + 720/20) * 18) + 6^4 - log_7(117649) + (368 mod 19) + floo... | {
"type": "arithmetic",
"seed": 1337,
"index": 7536
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. 637 - 459x = -632"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 637 - 459x = -632 $\n\nMove all terms to the left (standard form):\n$ -459x + 1269 = 0 $\n\nIsolate the $x$ term:\n$ -459x = -1269 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 7537
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Prove that $f(x,y)=\\frac12(6x^2+10y^2)+-1x+5y+-1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(6x^2+10y^2)+-1x+5y+-1 $\n\n$ \\frac{\\partial^2 f}{\... | {
"type": "optimization",
"seed": 1337,
"index": 7538
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. Find the modular inverse of 23 modulo 32, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $23\\cdot a^{-1} \\equiv 1 \\pmod{32}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarante... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7539
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Solve for $x$ (in radians): cos(x) = -\\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -\\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\a... | {
"type": "trigonometry",
"seed": 1337,
"index": 7540
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Prove that $f(x)=8x^2+5x+0$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+5x+0$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) ... | {
"type": "optimization",
"seed": 1337,
"index": 7541
} |
[
{
"role": "user",
"content": "solve: \n\n Evaluate: ((-59 + 288/16) * 19) + 8^5 - log_4(256) + (-370 mod 4) + ceil(149/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-59 + 288/16) * 19) + 8^5 - log_4(256) + (-370 mod 4) + ceil(149/2) $\n\nCompute each component:\n$ 288/1... | {
"type": "arithmetic",
"seed": 1337,
"index": 7542
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Find the modular inverse of 27 modulo 32, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $27\\cdot a^{-1} \\equiv 1 \\pmod{32}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity gua... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7543
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Find $A^{-1}$ for $A=\\begin{pmatrix}8&6\\\\-4&3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 8 & 6 \\\\ -4 & 3 \\end{pmatrix} $\n\nCompute the dete... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7544
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\part... | {
"type": "optimization",
"seed": 1337,
"index": 7545
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. -15 - 8 - 28x^2 = 173x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -15 - 8 - 28x^2 = 173x $\n\nMove all terms to the left (standard form):\n$ -28x^2 + -173x + -23 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7546
} |
[
{
"role": "user",
"content": "solve: Prove that $f(x)=6x^2+6x+-6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=6x^2+6x+-6$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 12x + 6 $\n$ f''(x) = 12 $\n\nSin... | {
"type": "optimization",
"seed": 1337,
"index": 7547
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. Evaluate: ((109 + -836/19) * 17) + 4^5 - log_3(3) + (-461 mod 2) + round(-105/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((109 + -836/19) * 17) + 4^5 - log_3(3) + (-461 mod 2) + round(-105/... | {
"type": "arithmetic",
"seed": 1337,
"index": 7548
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. -848 - 185 = 580x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -848 - 185 = 580x $\n\nMove all terms to the left (standard form):\n$ -580x + -1033 = 0 $\n\nIsolate the $x$ term:\n$ -580x = 1033 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 7549
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. 22x^2 - 173 - 27 = -187x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 22x^2 - 173 - 27 = -187x $\n\nMove all terms to the left (standard form):\n$ 22x^2 + 187x + -200 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7550
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Differentiate with respect to $x$: $f(x) = \\ln(1x + -1)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(1x + -1) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{... | {
"type": "calculus",
"seed": 1337,
"index": 7551
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Evaluate: ((-31 + -286/11) * 6) + 4^2 - log_9(9) + (198 mod 7) + round(-195/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-31 + -286/11) * 6) + 4^2 - log_9(9) + (198 mod 7) + round(-195/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 7552
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Evaluate: ((122 + 91/13) * 11) + 10^5 - log_4(64) + (430 mod 9) + floor(-7/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((122 + 91/13) * 11) + 10^5 - log_4(64) + (430 mod 9) + floor(-7/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 7553
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. -111x + 118 = -636"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -111x + 118 = -636 $\n\nMove all terms to the left (standard form):\n$ -111x + 754 = 0 $\n\nIsolate the $x$ term:\n$ -111x = -754 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 7554
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. -120 - 262x = 738"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -120 - 262x = 738 $\n\nMove all terms to the left (standard form):\n$ -262x + -858 = 0 $\n\nIsolate the $x$ term:\n$ -262x = 858 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7555
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Prove that $(\\mathbb{Z}_19, +)$ is a group under addition modulo 19."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_19=\\{0,1,\\dots,18\\}$ with operation $+$ modulo $19$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7556
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}0&2&4\\\\\\-4&5&3\\\\\\5&4&5\\end{pmatrix},\\quad b=\\begin{pmatrix}-6\\\\\\-24\\\\\\11\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7557
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. -64 + 302x = 654"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -64 + 302x = 654 $\n\nMove all terms to the left (standard form):\n$ 302x + -718 = 0 $\n\nIsolate the $x$ term:\n$ 302x = 718 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 7558
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. -314 - 101x = -534"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -314 - 101x = -534 $\n\nMove all terms to the left (standard form):\n$ -101x + 220 = 0 $\n\nIsolate the $x$ term:\n$ -101x = -220 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 7559
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Compute the indefinite integral: $\\int 2e^{1x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2e^{1x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n$ \... | {
"type": "calculus",
"seed": 1337,
"index": 7560
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Compute the determinant: $A=\\begin{pmatrix}5&-9\\\\3&-7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}5&-9\\\\\\3&-7\\end{pmatrix} $\n\nUse $\\d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7561
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Differentiate with respect to $x$: $f(x) = 3x^3 + -4x^2 + 2x + 12$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 3x^3 + -4x^2 + 2x + 12 $\n\nApply the standard differentiation rules:\n$ f'(x) =... | {
"type": "calculus",
"seed": 1337,
"index": 7562
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}6&4\\\\\\-4&4\\end{pmatrix},\\quad b=\\begin{pmatrix}30\\\\\\-60\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}6&4\\\\\\-4&4\\end... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7563
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. 33 - 80x - 47 = 83x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 33 - 80x - 47 = 83x^2 $\n\nMove all terms to the left (standard form):\n$ -83x^2 + -80x + -14 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7564
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. Solve for $x$ (in radians): cos(x) = -\\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -\\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\al... | {
"type": "trigonometry",
"seed": 1337,
"index": 7565
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Evaluate: ((183 + 60/4) * 11) + 5^2 - log_10(100) + (231 mod 24) + round(-188/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((183 + 60/4) * 11) + 5^2 - log_10(100) + (231 mod 24) + round(-188/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 7566
} |
[
{
"role": "user",
"content": "Please calculate this equation. Compute the determinant: $A=\\begin{pmatrix}5&2\\\\-9&8\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}5&2\\\\\\-9&8\\end{pmatrix} $\n\nUse $\\det(A)=ad-... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7567
} |
[
{
"role": "user",
"content": "Solve the following math equation. Evaluate: ((-129 + 805/23) * 12) + 12^2 - log_5(125) + (-304 mod 17) + round(104/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-129 + 805/23) * 12) + 12^2 - log_5(125) + (-304 mod 17) + round(104/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 7568
} |
[
{
"role": "user",
"content": "Please calculate this equation. Find the modular inverse of 9 modulo 20, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $9\\cdot a^{-1} \\equiv 1 \\pmod{20}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees integers ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7569
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. -737x + 198 = 918"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -737x + 198 = 918 $\n\nMove all terms to the left (standard form):\n$ -737x + -720 = 0 $\n\nIsolate the $x$ term:\n$ -737x = 720 $\n\nDivide both sides... | {
"type": "linear_equation",
"seed": 1337,
"index": 7570
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Compute the indefinite integral: $\\int -4x^3 + 1x^2 + 18x + -19\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -4x^3 + 1x^2 + 18x + -19\\,dx $\n\nUse standard antiderivatives ... | {
"type": "calculus",
"seed": 1337,
"index": 7571
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Evaluate: ((120 + 312/8) * 13) + 8^4 - log_3(3) + (-92 mod 11) + round(-64/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((120 + 312/8) * 13) + 8^4 - log_3(3) + (-92 mod 11) + round(-64/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 7572
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Evaluate: ((126 + 378/21) * 8) + 7^2 - log_10(100000) + (473 mod 23) + round(-147/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((126 + 378/21) * 8) + 7^2 - log_10(100000) + (473 mod 23) + round(... | {
"type": "arithmetic",
"seed": 1337,
"index": 7573
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Evaluate: ((189 + -799/17) * 14) + 5^2 - log_5(5) + (-11 mod 21) + round(-106/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((189 + -799/17) * 14) + 5^2 - log_5(5) + (-11 mod 21) + round(-106/2) ... | {
"type": "arithmetic",
"seed": 1337,
"index": 7574
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. -381x + 523 = 927"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -381x + 523 = 927 $\n\nMove all terms to the left (standard form):\n$ -381x + -404 = 0 $\n\nIsolate the $x$ term:\n$ -381x = 404 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7575
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\... | {
"type": "optimization",
"seed": 1337,
"index": 7576
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Evaluate: ((-183 + -216/18) * 18) + 5^2 - log_6(36) + (223 mod 19) + floor(-3/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-183 + -216/18) * 18) + 5^2 - log_6(36) + (223 mod 19) + floor(-3/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 7577
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. -17 - 71x^2 + 43 = -43x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -17 - 71x^2 + 43 = -43x $\n\nMove all terms to the left (standard form):\n$ -71x^2 + 43x + 26 = 0 $\n\nApply the quadratic formula:\n$ x... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7578
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Solve for $x$ (in radians): cos(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\a... | {
"type": "trigonometry",
"seed": 1337,
"index": 7579
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. Prove that $(\\mathbb{Z}_3, +)$ is a group under addition modulo 3."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_3=\\{0,1,\\dots,2\\}$ with operation $+$ modulo $3$.\n\n**Closure:** If $a,b\\in\\ma... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7580
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Evaluate: ((41 + 360/20) * 18) + 9^5 - log_9(9) + (-241 mod 16) + round(-16/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((41 + 360/20) * 18) + 9^5 - log_9(9) + (-241 mod 16) + round(-16/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 7581
} |
[
{
"role": "user",
"content": "Can you compute this equation? Compute $(72+132) \\bmod 10$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (72+132) \\bmod 10 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\n$ ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7582
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Compute $(-46+-109) \\bmod 25$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-46+-109) \\bmod 25 $\n\nUse the rule that addition is compatible wit... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7583
} |
[
{
"role": "user",
"content": "Solve the equation shown below. Evaluate: ((-57 + -243/9) * 14) + 7^2 - log_10(10) + (-485 mod 26) + ceil(-24/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-57 + -243/9) * 14) + 7^2 - log_10(10) + (-485 mod 26) + ceil(-24/2) $\n\nCompute ea... | {
"type": "arithmetic",
"seed": 1337,
"index": 7584
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. -3x - 149 = -39"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -3x - 149 = -39 $\n\nMove all terms to the left (standard form):\n$ -3x + -110 = 0 $\n\nIsolate the $x$ term:\n$ -3x = 110 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 7585
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Prove that $f(x)=1x^2+-7x+9$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=1x^2+-7x+9$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) =... | {
"type": "optimization",
"seed": 1337,
"index": 7586
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Compute the indefinite integral: $\\int -4x^3 + 2x^2 + -6x + 8\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -4x^3 + 2x^2 + -6x + 8\\,dx $\n\nUse standard antiderivatives (and integrati... | {
"type": "calculus",
"seed": 1337,
"index": 7587
} |
[
{
"role": "user",
"content": "Solve this equation accurately. -37x^2 + 188 - 149x = 168"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -37x^2 + 188 - 149x = 168 $\n\nMove all terms to the left (standard form):\n$ -37x^2 + -149x + 20 = 0 $\n\nApply the quadratic formula:\n$ x = ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7588
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-4&1&-2\\\\\\1&-1&4\\\\\\2&-2&3\\end{pmatrix},\\quad b=\\begin{pmatrix}-6\\\\\\7\\\\\\4\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7589
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Solve for $x$ (in radians): tan(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 7590
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{... | {
"type": "optimization",
"seed": 1337,
"index": 7591
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Evaluate: ((127 + -192/4) * 15) + 3^4 - log_7(117649) + (-251 mod 8) + floor(11/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((127 + -192/4) * 15) + 3^4 - log_7(117649) + (-251 mod 8) + floor(11/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 7592
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Solve for $x$ (in radians): sin(x) = \\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{1}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\a... | {
"type": "trigonometry",
"seed": 1337,
"index": 7593
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. -182x + 524 = -118"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -182x + 524 = -118 $\n\nMove all terms to the left (standard form):\n$ -182x + 642 = 0 $\n\nIsolate the $x$ term:\n$ -182x = -642 $\n\nDivi... | {
"type": "linear_equation",
"seed": 1337,
"index": 7594
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, hence:... | {
"type": "optimization",
"seed": 1337,
"index": 7595
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}4&-9\\\\\\-2&7\\end{pmatrix},\\quad b=\\begin{pmatrix}-56\\\\\\38\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}4&-9\\\\\\-2&7\\en... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7596
} |
[
{
"role": "user",
"content": "solve: Compute the indefinite integral: $\\int 2x^3 + -7x^2 + -5x + 10\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2x^3 + -7x^2 + -5x + 10\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n$ \\... | {
"type": "calculus",
"seed": 1337,
"index": 7597
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? -29 - 175x^2 - 84 = 160x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -29 - 175x^2 - 84 = 160x $\n\nMove all terms to the left (standard form):\n$ -175x^2 + -160x + -113 = 0 $\n\nApply the quadrati... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7598
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Prove that $f(x)=2x^2+-6x+9$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=2x^2+-6x+9$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ ... | {
"type": "optimization",
"seed": 1337,
"index": 7599
} |
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