question stringlengths 12 86 | problem stringclasses 31
values | how_to_solve stringlengths 8 96 | answer stringlengths 1 24 |
|---|---|---|---|
Solve the system:
-2x + -3y = -14
4x + -4y = -12 | Find integer x and y that satisfy both equations. | Use substitution or elimination. | x=1, y=4 |
Solve for x: (-10/11)x + 2 = -108 | Linear equation with fractional coefficient. | Subtract c then multiply by b/a (watch sign). | 121 |
Simplify: -4x + -5x + -4y | Combine like terms. | Add coefficients of like terms (x with x, y with y). | -9x - 4y |
What is 96% of 100? | Compute the percent value (integer-safe). | Compute 96/100 × 100. | 96 |
Find one integer solution to 8x + 9y = -12 | Linear Diophantine equation (return one integer pair). | Find any particular solution (here constructed). | x=3, y=-4 |
What is 95 mod 10 (remainder after division)? | Compute remainder when a is divided by b. | Divide 95 by 10 and take the integer remainder 0..9. | 5 |
What is -64 mod 18 (remainder after division)? | Compute remainder when a is divided by b. | Divide -64 by 18 and take the integer remainder 0..17. | 8 |
What is -16% of 225? | Compute the percent value (integer-safe). | Compute -16/100 × 225. | -36 |
What is 122 mod 26 (remainder after division)? | Compute remainder when a is divided by b. | Divide 122 by 26 and take the integer remainder 0..25. | 18 |
Solve for x: (2x + 100)/(3x + -10) = -2 | Rational equation that reduces to a linear equation. | Cross-multiply to get linear equation and solve for x (ensure denominator ≠ 0). | -10 |
Solve: 10x + -4 ≤ -34 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x ≤ -3 |
Solve for x: 1x^3 + -5x^2 + 3x + 9 = 0 | Cubic polynomial factorable to (x - r1)(x - r2)(x - r3). | Find integer root by rational root theorem then factor. | -1, 3, 3 |
Factor the polynomial: 1x^2 + -4x + -5 | Factor quadratic into integer linear factors. | Find integers r1 and r2 such that polynomial = (x - r1)(x - r2). | (x - -1)(x - 5) |
What is 1 - 259? | Subtract the second number from the first. | Subtract 259 from 1. | -258 |
Solve for x: 1x^2 - 1x - 6 = 0 | Solve the quadratic equation with integer roots. | Factor or use quadratic formula (should factor cleanly). | -2, 3 |
What is -47 mod 5 (remainder after division)? | Compute remainder when a is divided by b. | Divide -47 by 5 and take the integer remainder 0..4. | 3 |
What is 391 ÷ -23? | Divide evenly (negative allowed). | Divide 391 by -23. | -17 |
Solve for x: (8x + -118)/(-1x + -9) = 2 | Rational equation that reduces to a linear equation. | Cross-multiply to get linear equation and solve for x (ensure denominator ≠ 0). | 10 |
Solve for x: (8/11)x + -8 = 72 | Linear equation with fractional coefficient. | Subtract c then multiply by b/a (watch sign). | 110 |
Solve: 10x + 10 ≥ 50 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x ≥ 4 |
What is 131 - 109? | Subtract the second number from the first. | Subtract 109 from 131. | 22 |
Solve for x: sqrt(-2x + 24) = 4 | Radical equation; return integer solution if it exists. | Square both sides to get ax + b = c^2, then solve. | 4 |
What is -2^2? | Evaluate the integer exponent. | Multiply -2 by itself 2 times (0th power = 1). | 4 |
What is -378 ÷ 27? | Divide evenly. | Divide -378 by 27. | -14 |
Convert the improper fraction 91/5 to a mixed number. | Convert to mixed number and simplify the fractional part. | Divide numerator by denominator to get whole part and remainder; simplify remainder/denominator. | 18 1/5 |
Simplify: -2x + -5x + 8y | Combine like terms. | Add coefficients of like terms (x with x, y with y). | -7x + 8y |
What is -3^3? | Evaluate the integer exponent. | Multiply -3 by itself 3 times (0th power = 1). | -27 |
What is -17 × 26? | Multiply the numbers. | Multiply -17 by 26. | -442 |
Solve the system:
-4x + -3y = 34
-2x + 1y = 2 | Find integer x and y that satisfy both equations. | Use substitution or elimination. | x=-4, y=-6 |
Solve for x: (1/7)x + 10 = 19 | Linear equation with fractional coefficient. | Subtract c then multiply by b/a (watch sign). | 63 |
Solve for x: 1x^2 - 10x + 16 = 0 | Solve the quadratic equation with integer roots. | Factor or use quadratic formula (should factor cleanly). | 2, 8 |
Solve for x: 5^x = 5 | Solve the exponential equation for integer x. | Find x such that repeated multiplication of 5 yields 5. | 1 |
Solve for x: -6(x + 8) = -42 | Solve a two-step equation. | Divide both sides by a, then subtract b. | -1 |
What is -292 - -65? | Subtract the second number from the first. | Subtract -65 from -292. | -227 |
Solve the system:
4x + 2y + 2z = 6
1x + 0y + -4z = -23
3x + -4y + 2z = -15 | Find integer x,y,z satisfying the linear system. | Use elimination or Cramer's rule (matrix is invertible). | x=-3, y=4, z=5 |
What is -20 × -7? | Multiply the numbers. | Multiply -20 by -7. | 140 |
Factor the polynomial: 1x^2 + -6x + 0 | Factor quadratic into integer linear factors. | Find integers r1 and r2 such that polynomial = (x - r1)(x - r2). | (x - 0)(x - 6) |
Solve for x: log base 2 of x = 1 | Compute x from the log equation (integer-safe). | x = 2^1. | 2 |
What is 16 × -9? | Multiply the numbers. | Multiply 16 by -9. | -144 |
Solve the system:
-3x + 3y = -9
5x + -2y = 33 | Find integer x and y that satisfy both equations. | Use substitution or elimination. | x=9, y=6 |
Convert the improper fraction 182/9 to a mixed number. | Convert to mixed number and simplify the fractional part. | Divide numerator by denominator to get whole part and remainder; simplify remainder/denominator. | 20 2/9 |
Solve for x: 1x^3 + -5x^2 + 3x + 9 = 0 | Cubic polynomial factorable to (x - r1)(x - r2)(x - r3). | Find integer root by rational root theorem then factor. | -1, 3, 3 |
Solve: |2x + -11| = 19 | Find integer solutions x, if any. | Set inside equal to ±c and solve both linear equations. | 15, -4 |
Solve: -10x + -19 > -189 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x > 17 |
What is 110% of 90? | Compute the percent value (integer-safe). | Compute 110/100 × 90. | 99 |
What is 145% of 100? | Compute the percent value (integer-safe). | Compute 145/100 × 100. | 145 |
Solve for x: 1x^3 + 4x^2 + 1x + -6 = 0 | Cubic polynomial factorable to (x - r1)(x - r2)(x - r3). | Find integer root by rational root theorem then factor. | -3, -2, 1 |
Factor the polynomial: x^3 - 1x^2 + -25x - -25 | Factor cubic into linear integer factors if possible. | Find integer roots and factor as (x - r1)(x - r2)(x - r3). | (x - 1)(x - -5)(x - 5) |
Solve for x: sqrt(5x + 115) = 10 | Radical equation; return integer solution if it exists. | Square both sides to get ax + b = c^2, then solve. | -3 |
Simplify -140/14 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | -10 |
What is -27 ÷ 27? | Divide evenly. | Divide -27 by 27. | -1 |
If each item costs -2 dollars and the total cost is 24, how many items were bought? | Solve using algebra. | Divide total cost by price per item. | -12 |
What is -37 × 11? | Multiply the numbers. | Multiply -37 by 11. | -407 |
Solve for x: 1x^3 + 0x^2 + -4x + 0 = 0 | Cubic polynomial factorable to (x - r1)(x - r2)(x - r3). | Find integer root by rational root theorem then factor. | -2, 0, 2 |
What is -52 × 58? | Multiply the numbers. | Multiply -52 by 58. | -3016 |
What is -174 mod 20 (remainder after division)? | Compute remainder when a is divided by b. | Divide -174 by 20 and take the integer remainder 0..19. | 6 |
What is 39 + -225? | Add the two numbers. | Add 39 and -225. | -186 |
What is 25 × 58? | Multiply the numbers. | Multiply 25 by 58. | 1450 |
Solve: |-2x + 15| = 14 | Find integer solutions x, if any. | Set inside equal to ±c and solve both linear equations. | no integer solutions |
Solve for x: 2x^2 + 18x + 36 = 0 | Solve the quadratic equation with integer roots. | Factor or use quadratic formula (should factor cleanly). | -3, -6 |
Solve for x: 3^x = 243 | Solve the exponential equation for integer x. | Find x such that repeated multiplication of 3 yields 243. | 5 |
Solve for x: -2x^2 - 10x + 12 = 0 | Solve the quadratic equation with integer roots. | Factor or use quadratic formula (should factor cleanly). | 1, -6 |
Find smallest non-negative x such that -9x ≡ 13 (mod 22) | Solve linear congruence. | Use gcd and modular inverse on reduced modulus (if solution exists). | 1 |
Solve for x: -11x + -8 = 58 | Solve the linear equation. | Subtract b from both sides, then divide by a. | -6 |
Solve: |5x + -2| ≤ 12 | Find integer range of x satisfying the absolute inequality. | Isolate and produce interval then take integer bounds. | -2 ≤ x ≤ 2 |
What is 144 mod 4 (remainder after division)? | Compute remainder when a is divided by b. | Divide 144 by 4 and take the integer remainder 0..3. | 0 |
Solve for x: (-6x + 29)/(3x + -5) = -1 | Rational equation that reduces to a linear equation. | Cross-multiply to get linear equation and solve for x (ensure denominator ≠ 0). | 8 |
Factor the polynomial: x^3 - 10x^2 + 31x - 30 | Factor cubic into linear integer factors if possible. | Find integer roots and factor as (x - r1)(x - r2)(x - r3). | (x - 5)(x - 2)(x - 3) |
Solve for x: 2x^2 + 8x + 6 = 0 | Solve the quadratic equation with integer roots. | Factor or use quadratic formula (should factor cleanly). | -1, -3 |
Factor the polynomial: 1x^2 + 5x + 4 | Factor quadratic into integer linear factors. | Find integers r1 and r2 such that polynomial = (x - r1)(x - r2). | (x - -1)(x - -4) |
Convert the improper fraction 19/3 to a mixed number. | Convert to mixed number and simplify the fractional part. | Divide numerator by denominator to get whole part and remainder; simplify remainder/denominator. | 6 1/3 |
Solve for x: 5^x = 125 | Solve the exponential equation for integer x. | Find x such that repeated multiplication of 5 yields 125. | 3 |
Find smallest non-negative x such that -8x ≡ 0 (mod 4) | Solve linear congruence. | Use gcd and modular inverse on reduced modulus (if solution exists). | 0 |
Solve for x: 4x^2 + 32x + 64 = 0 | Quadratic with integer roots and non-unit leading coefficient. | Factor or use quadratic formula; roots are integers. | -4, -4 |
If each item costs -3 dollars and the total cost is -15, how many items were bought? | Solve using algebra. | Divide total cost by price per item. | 5 |
Solve for x: sqrt(-4x + 24) = 2 | Radical equation; return integer solution if it exists. | Square both sides to get ax + b = c^2, then solve. | 5 |
What is -27 ÷ -9? | Divide evenly. | Divide -27 by -9. | 3 |
Solve for x: -3(x + -11) = 69 | Solve a two-step equation. | Divide both sides by a, then subtract b. | -12 |
Solve for x: sqrt(2x + 20) = 0 | Radical equation; return integer solution if it exists. | Square both sides to get ax + b = c^2, then solve. | -10 |
Solve: |-1x + 15| = 13 | Find integer solutions x, if any. | Set inside equal to ±c and solve both linear equations. | 2, 28 |
Solve: 1x + 2 < -13 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x < -15 |
Solve for x: 8(x + -12) = -144 | Solve a two-step equation. | Divide both sides by a, then subtract b. | -6 |
Solve for x: -4x^2 + -24x + -20 = 0 | Quadratic with integer roots and non-unit leading coefficient. | Factor or use quadratic formula; roots are integers. | -5, -1 |
Solve for x: log base 2 of x = 0 | Compute x from the log equation (integer-safe). | x = 2^0. | 1 |
Solve for x: 1x^2 + 11x + 30 = 0 | Solve the quadratic equation with integer roots. | Factor or use quadratic formula (should factor cleanly). | -6, -5 |
Solve for x: (-1x + 0)/(2x + 3) = 1 | Rational equation that reduces to a linear equation. | Cross-multiply to get linear equation and solve for x (ensure denominator ≠ 0). | -1 |
Solve for x: 3^x = 3 | Solve the exponential equation for integer x. | Find x such that repeated multiplication of 3 yields 3. | 1 |
What is -62 + 203? | Add the two numbers. | Add -62 and 203. | 141 |
Solve for x: 1x^3 + -3x^2 + -9x + -5 = 0 | Cubic polynomial factorable to (x - r1)(x - r2)(x - r3). | Find integer root by rational root theorem then factor. | -1, -1, 5 |
Solve for x: 4^x = 64 | Solve the exponential equation for integer x. | Find x such that repeated multiplication of 4 yields 64. | 3 |
Solve for x: 5x + 19 = -46 | Solve the linear equation. | Subtract b from both sides, then divide by a. | -13 |
Solve: |-5x + -4| = 1 | Find integer solutions x, if any. | Set inside equal to ±c and solve both linear equations. | -1 |
Solve the system:
4x + 1y = 19
-1x + 5y = -10 | Find integer x and y that satisfy both equations. | Use substitution or elimination. | x=5, y=-1 |
Solve for x: 3x^2 + -15x + -42 = 0 | Quadratic with integer roots and non-unit leading coefficient. | Factor or use quadratic formula; roots are integers. | 7, -2 |
Find one integer solution to 5x + 6y = 49 | Linear Diophantine equation (return one integer pair). | Find any particular solution (here constructed). | x=5, y=4 |
Solve for x: log base 3 of x = 0 | Compute x from the log equation (integer-safe). | x = 3^0. | 1 |
Solve: |2x + 11| ≤ 4 | Find integer range of x satisfying the absolute inequality. | Isolate and produce interval then take integer bounds. | -7 ≤ x ≤ -4 |
Factor the polynomial: 1x^2 + -9x + 20 | Factor quadratic into integer linear factors. | Find integers r1 and r2 such that polynomial = (x - r1)(x - r2). | (x - 4)(x - 5) |
Solve for x: -3(x + 5) = -18 | Solve a two-step equation. | Divide both sides by a, then subtract b. | 1 |
What is 68% of 150? | Compute the percent value (integer-safe). | Compute 68/100 × 150. | 102 |
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