messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Please find and report the solution to this equation. -60 - 953x = 669"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -60 - 953x = 669 $\n\nMove all terms to the left (standard form):\n$ -953x + -729 = 0 $\n\nIsolate the $x$ term:\n$ -953x = 729 $\n\n... | {
"type": "linear_equation",
"seed": 1337,
"index": 1200
} |
[
{
"role": "user",
"content": "Solve the equation shown below. Evaluate: ((-143 + 432/9) * 16) + 3^2 - log_2(16) + (328 mod 22) + floor(-99/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-143 + 432/9) * 16) + 3^2 - log_2(16) + (328 mod 22) + floor(-99/2) $\n\nCompute each... | {
"type": "arithmetic",
"seed": 1337,
"index": 1201
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-6&1\\\\\\-5&-7\\end{pmatrix},\\quad b=\\begin{pmatrix}-59\\\\\\-10\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-6&1\\\\\\-5&... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1202
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. 667 - 844x = -518"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 667 - 844x = -518 $\n\nMove all terms to the left (standard form):\n$ -844x + 1185 = 0 $\n\nIsolate the $x$ term:\n$ -844x = -1185 $\n\nDi... | {
"type": "linear_equation",
"seed": 1337,
"index": 1203
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. 9 - 88 + 177x^2 = 6x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 9 - 88 + 177x^2 = 6x $\n\nMove all terms to the left (standard form):\n$ 177x^2 + -6x + -79 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 1204
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. -15x + 31 - 53x^2 = 173"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -15x + 31 - 53x^2 = 173 $\n\nMove all terms to the left (standard form):\n$ -53x^2 + -15x + -142 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 1205
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Prove that $f(x)=9x^2+0x+-10$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=9x^2+0x+-10$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 1... | {
"type": "optimization",
"seed": 1337,
"index": 1206
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Prove that $f(x)=3x^2+-1x+-7$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=3x^2+-1x+-7$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(... | {
"type": "optimization",
"seed": 1337,
"index": 1207
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. Compute $(-151+-98) \\bmod 15$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-151+-98) \\bmod 15 $\n\nUse the rule that addition is compatible with reduction modulo... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 1208
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. -859 - 864x = 788"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -859 - 864x = 788 $\n\nMove all terms to the left (standard form):\n$ -864x + -1647 = 0 $\n\nIsolate the $x$ term:\n$ -864x = 1647 $\n\nDivi... | {
"type": "linear_equation",
"seed": 1337,
"index": 1209
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-3&4\\\\\\3&7\\end{pmatrix},\\quad b=\\begin{pmatrix}-24\\\\\\-75\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-3&4\\\\\\3&... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1210
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}3&-3&-5\\\\\\-5&-2&-3\\\\\\-2&4&0\\end{pmatrix},\\quad b=\\begin{pmatrix}9\\\\\\33\\\\\\0\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g.... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1211
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Find $A^{-1}$ for $A=\\begin{pmatrix}2&5\\\\9&-2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 2 & 5 \\\\ 9 & -2 \\end{pmatrix} $\n\nCompute the dete... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1212
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Prove that $(\\mathbb{Z}_17, +)$ is a group under addition modulo 17."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_17=\\{0,1,\\dots,16\\}$ with operation $+$ modulo $17$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 1213
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. 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) = 3.14159 $\n\... | {
"type": "trigonometry",
"seed": 1337,
"index": 1214
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. 586x - 6 = -122"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 586x - 6 = -122 $\n\nMove all terms to the left (standard form):\n$ 586x + 116 = 0 $\n\nIsolate the $x$ term:\n$ 586x = -116 $\n\nDivide both... | {
"type": "linear_equation",
"seed": 1337,
"index": 1215
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Prove that $(\\mathbb{Z}_21, +)$ is a group under addition modulo 21."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_21=\\{0,1,\\dots,20\\}$ with operation $+$ modulo $21$.\n\n**Closure:** If $a,b\\in\\mathb... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 1216
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. 274x - 368 = 162"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 274x - 368 = 162 $\n\nMove all terms to the left (standard form):\n$ 274x + -530 = 0 $\n\nIsolate the $x$ term:\n$ 274x = 530 $\n\nDivide both... | {
"type": "linear_equation",
"seed": 1337,
"index": 1217
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Find the modular inverse of 17 modulo 24, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $17\\cdot a^{-1} \\equiv 1 \\pmod{24}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarant... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 1218
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Compute the determinant: $A=\\begin{pmatrix}6&4\\\\6&-2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}6&4\\\\\\6&-2\\end{pmatrix} $\n\nUse $\\det(A)=ad-bc$: \n$ \\det(A... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1219
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Evaluate: ((56 + -56/8) * 10) + 3^2 - log_9(59049) + (341 mod 15) + round(-29/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((56 + -56/8) * 10) + 3^2 - log_9(59049) + (341 mod 15) + round(-29/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 1220
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. 463 - 239x = 870"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 463 - 239x = 870 $\n\nMove all terms to the left (standard form):\n$ -239x + -407 = 0 $\n\nIsolate the $x$ term:\n$ -239x = 407 $\n\nDivide both si... | {
"type": "linear_equation",
"seed": 1337,
"index": 1221
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Solve for $x$ (in radians): sin(x) = \\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha... | {
"type": "trigonometry",
"seed": 1337,
"index": 1222
} |
[
{
"role": "user",
"content": "Solve this equation accurately. x + 3 - 43x^2 = -175"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ x + 3 - 43x^2 = -175 $\n\nMove all terms to the left (standard form):\n$ -43x^2 + 1x + 178 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b \... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 1223
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. -46 + 44 + 117x = -64x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -46 + 44 + 117x = -64x^2 $\n\nMove all terms to the left (standard form):\n$ 64x^2 + 117x + -2 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 1224
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Solve for $x$ (in radians): sin(x) = -\\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = -\\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alph... | {
"type": "trigonometry",
"seed": 1337,
"index": 1225
} |
[
{
"role": "user",
"content": "Determine the value produced by 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) = 3.14159 $\... | {
"type": "trigonometry",
"seed": 1337,
"index": 1226
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. -10x^2 + 31 - 134 = -6x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -10x^2 + 31 - 134 = -6x $\n\nMove all terms to the left (standard form):\n$ -10x^2 + 6x + -103 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 1227
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. Evaluate: ((-21 + -60/5) * 7) + 2^2 - log_2(16) + (-74 mod 21) + ceil(-157/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-21 + -60/5) * 7) + 2^2 - log_2(16) + (-74 mod 21) + ceil(-157/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 1228
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. 37 + 88 + 61x^2 = -127x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 37 + 88 + 61x^2 = -127x $\n\nMove all terms to the left (standard form):\n$ 61x^2 + 127x + 125 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 1229
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. 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$ \... | {
"type": "trigonometry",
"seed": 1337,
"index": 1230
} |
[
{
"role": "user",
"content": "Calculate the result of the given 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... | {
"type": "trigonometry",
"seed": 1337,
"index": 1231
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Differentiate with respect to $x$: $f(x) = 1x^3 + 7x^2 + -17x + 1$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 1x^3 + 7x^2 + -17x + 1 $\n\nApply the standard differentiation rules:\n$ f'... | {
"type": "calculus",
"seed": 1337,
"index": 1232
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Evaluate: ((178 + 42/7) * 17) + 2^4 - log_2(8) + (398 mod 4) + floor(-83/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((178 + 42/7) * 17) + 2^4 - log_2(8) + (398 mod 4) + floor(-83/2) $\n\nCompute ... | {
"type": "arithmetic",
"seed": 1337,
"index": 1233
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. 46x - 179 + 51 = -154x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 46x - 179 + 51 = -154x^2 $\n\nMove all terms to the left (standard form):\n$ 154x^2 + 46x + -128 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 1234
} |
[
{
"role": "user",
"content": "Please calculate this equation. Prove that $f(x)=4x^2+-10x+2$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=4x^2+-10x+2$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 8x + -1... | {
"type": "optimization",
"seed": 1337,
"index": 1235
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. 26 - 195x - 72 = -103x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 26 - 195x - 72 = -103x^2 $\n\nMove all terms to the left (standard form):\n$ 103x^2 + -195x + -46 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 1236
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}2&7\\\\\\-6&0\\end{pmatrix},\\quad b=\\begin{pmatrix}52\\\\\\12\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}2&7\\\\\\-6&... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1237
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Compute the determinant: $A=\\begin{pmatrix}-6&5\\\\9&-5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-6&5\\\\\\9&-5\\end{pmatrix} $\n\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1238
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. 234 - 983x = -766"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 234 - 983x = -766 $\n\nMove all terms to the left (standard form):\n$ -983x + 1000 = 0 $\n\nIsolate the $x$ term:\n$ -983x = -1000 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 1239
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Prove that $f(x,y)=\\frac12(2x^2+4y^2)+6x+-1y+-8$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(2x^2+4y^2)+6x+-1y+-8 $\n\n$ \\frac{\\partial^2 f}... | {
"type": "optimization",
"seed": 1337,
"index": 1240
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. -258 - 662 = -486x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -258 - 662 = -486x $\n\nMove all terms to the left (standard form):\n$ 486x + -920 = 0 $\n\nIsolate the $x$ term:\n$ 486x = 920 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 1241
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Find $A^{-1}$ for $A=\\begin{pmatrix}3&6\\\\0&3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 3 & 6 \\\\ 0 & 3 \\end{pmatrix} $\n\nCompute th... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1242
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-2&-2&1\\\\\\-4&2&5\\\\\\-3&-3&3\\end{pmatrix},\\quad b=\\begin{pmatrix}-4\\\\\\10\\\\\\-3\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1243
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. 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$ \\al... | {
"type": "trigonometry",
"seed": 1337,
"index": 1244
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. 43x^2 + 0x - 150 = -115"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 43x^2 + 0x - 150 = -115 $\n\nMove all terms to the left (standard form):\n$ 43x^2 + 0x + -35 = 0 $\n\nApply the quadratic f... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 1245
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Compute the determinant: $A=\\begin{pmatrix}6&-2\\\\-9&7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}6&-2\\\\\\-9&7\\end{pmatrix} $\n\nU... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1246
} |
[
{
"role": "user",
"content": "solve: \n\n Compute the indefinite integral: $\\int -3x^3 + 5x^2 + -4x + 11\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -3x^3 + 5x^2 + -4x + 11\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n... | {
"type": "calculus",
"seed": 1337,
"index": 1247
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Compute the indefinite integral: $\\int -9\\sin(2x) + -6\\cos(8x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -9\\sin(2x) + -6\\cos(8x)\\,dx $\n\nUse standard antiderivatives (a... | {
"type": "calculus",
"seed": 1337,
"index": 1248
} |
[
{
"role": "user",
"content": "Evaluate the following 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) = 3.14159 $\n\nThe general... | {
"type": "trigonometry",
"seed": 1337,
"index": 1249
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Solve for $x$ (in radians): tan(x) = -\\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = -\\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\... | {
"type": "trigonometry",
"seed": 1337,
"index": 1250
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Prove that $f(x,y)=\\frac12(1x^2+3y^2)+-6x+-6y+-1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+3y^2)+-6x+-6y+-1 $\n\n$ \\frac{\\partial^2 f}{\\partia... | {
"type": "optimization",
"seed": 1337,
"index": 1251
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Solve for $x$ (in radians): tan(x) = -\\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = -\\frac{1}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(-\\... | {
"type": "trigonometry",
"seed": 1337,
"index": 1252
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. 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": 1253
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. 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 $\n\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 1254
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. -994 + 504 = 35x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -994 + 504 = 35x $\n\nMove all terms to the left (standard form):\n$ -35x + -490 = 0 $\n\nIsolate the $x$ term:\n$ -35x = 490 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 1255
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Compute the indefinite integral: $\\int -1e^{3x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -1e^{3x}\\,dx $\n\nUse standard antiderivatives (and integration by parts ... | {
"type": "calculus",
"seed": 1337,
"index": 1256
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. 315 - 876x = 222"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 315 - 876x = 222 $\n\nMove all terms to the left (standard form):\n$ -876x + 93 = 0 $\n\nIsolate the $x$ term:\n$ -876x = -93 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 1257
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Compute the determinant: $A=\\begin{pmatrix}-9&-6\\\\-1&-9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-9&-6\\\\\\-1&-9\\end{pmatrix} $\n\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1258
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Evaluate: ((18 + 252/6) * 7) + 2^3 - log_6(46656) + (-14 mod 17) + floor(-109/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((18 + 252/6) * 7) + 2^3 - log_6(46656) + (-14 mod 17) + floor(-10... | {
"type": "arithmetic",
"seed": 1337,
"index": 1259
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. 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": 1260
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. -27x - 172 - 74 = -26x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -27x - 172 - 74 = -26x^2 $\n\nMove all terms to the left (standard form):\n$ 26x^2 + -27x + -246 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 1261
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Compute the determinant: $A=\\begin{pmatrix}-6&0\\\\5&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-6&0\\\\\\5&6\\end{pmatrix} $\n\nUs... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1262
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. Evaluate: ((-132 + -351/9) * 8) + 11^4 - log_9(531441) + (367 mod 16) + floor(-8/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-132 + -351/9) * 8) + 11^4 - log_9(531441) + (367 mod 16) + floor(-8/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 1263
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}4&7\\\\-9&7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 4 & 7 \\\\ -9 & 7 \\end{pmatrix} $\n\nCompute the deter... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1264
} |
[
{
"role": "user",
"content": "Resolve the equation provided. -465 + 7x = 680"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -465 + 7x = 680 $\n\nMove all terms to the left (standard form):\n$ 7x + -1145 = 0 $\n\nIsolate the $x$ term:\n$ 7x = 1145 $\n\nDivide both sides by the coef... | {
"type": "linear_equation",
"seed": 1337,
"index": 1265
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}5&-2\\\\\\-3&1\\end{pmatrix},\\quad b=\\begin{pmatrix}39\\\\\\-23\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}5&-2\\\\\\-3&1\\en... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1266
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. 9 - 26x^2 - 153x = 162"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 9 - 26x^2 - 153x = 162 $\n\nMove all terms to the left (standard form):\n$ -26x^2 + -153x + -153 = 0 $\n\nApply the quadrat... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 1267
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Solve for $x$ (in radians): sin(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ ... | {
"type": "trigonometry",
"seed": 1337,
"index": 1268
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. -91 + 224 = 713x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -91 + 224 = 713x $\n\nMove all terms to the left (standard form):\n$ -713x + 133 = 0 $\n\nIsolate the $x$ term:\n$ -713x = -133 $\n\nDivide both sid... | {
"type": "linear_equation",
"seed": 1337,
"index": 1269
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. Compute the indefinite integral: $\\int \\ln(2x + -2)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(2x + -2)\\,dx $\n\nUse standard antiderivatives (and integration by parts if ne... | {
"type": "calculus",
"seed": 1337,
"index": 1270
} |
[
{
"role": "user",
"content": "Please solve this math problem. -27x + 38 + 140x^2 = -197"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -27x + 38 + 140x^2 = -197 $\n\nMove all terms to the left (standard form):\n$ 140x^2 + -27x + 235 = 0 $\n\nApply the quadratic formula:\n$ x = ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 1271
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Evaluate: ((-4 + 340/17) * 6) + 2^3 - log_8(64) + (-393 mod 18) + ceil(-28/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-4 + 340/17) * 6) + 2^3 - log_8(64) + (-393 mod 18) + ceil(-28/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 1272
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Solve for $x$ (in radians): tan(x) = \\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = \\frac{1}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(... | {
"type": "trigonometry",
"seed": 1337,
"index": 1273
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Differentiate with respect to $x$: $f(x) = -8e^{4x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -8e^{4x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-32e^{4x}} $"
}
... | {
"type": "calculus",
"seed": 1337,
"index": 1274
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. 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$... | {
"type": "trigonometry",
"seed": 1337,
"index": 1275
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Evaluate: ((-130 + -390/15) * 6) + 12^2 - log_7(2401) + (-362 mod 28) + floor(103/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-130 + -390/15) * 6) + 12^2 - log_7(2401) + (-362 mod 28) ... | {
"type": "arithmetic",
"seed": 1337,
"index": 1276
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Prove that $(\\mathbb{Z}_28, +)$ is a group under addition modulo 28."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_28=\\{0,1,\\dots,27\\}$ with operation $+$ modulo $28$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, then... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 1277
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. -40 + 148x + 54x^2 = 103"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -40 + 148x + 54x^2 = 103 $\n\nMove all terms to the left (standard form):\n$ 54x^2 + 148x + -143 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 1278
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. Compute the indefinite integral: $\\int -8\\sin(4x) + -9\\cos(6x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -8\\sin(4x) + -9\\cos(6x)\\,dx $\n\nUse standard antiderivatives (a... | {
"type": "calculus",
"seed": 1337,
"index": 1279
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. 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": 1280
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. 749 - 954 = 493x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 749 - 954 = 493x $\n\nMove all terms to the left (standard form):\n$ -493x + -205 = 0 $\n\nIsolate the $x$ term:\n$ -493x = 205 $\n\nDivi... | {
"type": "linear_equation",
"seed": 1337,
"index": 1281
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Compute $(82+-105) \\bmod 17$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (82+-105) \\bmod 17 $\n\nUse the rule that addition is compatible with reduction mod... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 1282
} |
[
{
"role": "user",
"content": "Can you compute this equation? 557 + 513x = 604"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 557 + 513x = 604 $\n\nMove all terms to the left (standard form):\n$ 513x + -47 = 0 $\n\nIsolate the $x$ term:\n$ 513x = 47 $\n\nDivide both sides by the co... | {
"type": "linear_equation",
"seed": 1337,
"index": 1283
} |
[
{
"role": "user",
"content": "Solve the following math equation. Differentiate with respect to $x$: $f(x) = \\ln(5x + 2)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(5x + 2) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{5}{5x + 2}... | {
"type": "calculus",
"seed": 1337,
"index": 1284
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. Find $A^{-1}$ for $A=\\begin{pmatrix}8&8\\\\-3&1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 8 & 8 \\\\ -3 & 1 \\end{pmatrix} $\n\nCompute t... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1285
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Compute $(-132+-142) \\bmod 21$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-132+-142) \\bmod 21 $\n\nUse the rule that addition is compatible with reduction... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 1286
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. 101 + 878 = -335x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 101 + 878 = -335x $\n\nMove all terms to the left (standard form):\n$ 335x + 979 = 0 $\n\nIsolate the $x$ term:\n$ 335x = -979 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 1287
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-4&2&-1\\\\\\-2&3&1\\\\\\-5&-2&1\\end{pmatrix},\\quad b=\\begin{pmatrix}-18\\\\\\-1\\\\\\-18\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1288
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}2&3&0\\\\\\4&5&2\\\\\\3&-1&-2\\end{pmatrix},\\quad b=\\begin{pmatrix}-1\\\\\\3\\\\\\13\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1289
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Prove that $f(x,y)=\\frac12(8x^2+10y^2)+-1x+-4y+-5$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(8x^2+10y^2)+-1x+-4y+-5 $\n\n$ \\frac{\\part... | {
"type": "optimization",
"seed": 1337,
"index": 1290
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. 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{\\par... | {
"type": "optimization",
"seed": 1337,
"index": 1291
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Find $A^{-1}$ for $A=\\begin{pmatrix}2&1\\\\9&7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 2 & 1 \\\\ 9 & 7 \\end{pmatrix} $\n\nCompute the det... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1292
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Evaluate: ((-108 + -517/11) * 18) + 11^5 - log_7(16807) + (314 mod 12) + ceil(103/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-108 + -517/11) * 18) + 11^5 - log_7(16807) + (314 mod 12) + ceil(103/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 1293
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Compute $(-190+-135) \\bmod 14$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-190+-135) \\bmod 14 $\n\nUse the rule that addition is compatible with reduction ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 1294
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Compute the determinant: $A=\\begin{pmatrix}7&-9\\\\2&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}7&-9\\\\\\2&5\\end{pmatrix} $\n\nUse $... | {
"type": "linear_algebra",
"seed": 1337,
"index": 1295
} |
[
{
"role": "user",
"content": "Please solve this math problem. Evaluate: ((59 + -84/14) * 9) + 8^4 - log_3(729) + (-348 mod 28) + floor(-22/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((59 + -84/14) * 9) + 8^4 - log_3(729) + (-348 mod 28) + floor(-22/2) $\n\nCompute each... | {
"type": "arithmetic",
"seed": 1337,
"index": 1296
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. -478 - 975x = -671"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -478 - 975x = -671 $\n\nMove all terms to the left (standard form):\n$ -975x + 193 = 0 $\n\nIsolate the $x$ term:\n$ -975x = -193 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 1297
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. Solve for $x$ (in radians): sin(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 1298
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Compute $(-161+-68) \\bmod 5$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-161+-68) \\bmod 5 $\n\nUse the rule that addition is compatible with reduction mod... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 1299
} |
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