question stringlengths 12 86 | problem stringclasses 31 values | how_to_solve stringlengths 8 96 | answer stringlengths 1 24 |
|---|---|---|---|
What is -3^2? | Evaluate the integer exponent. | Multiply -3 by itself 2 times (0th power = 1). | 9 |
Factor the polynomial: x^3 - 11x^2 + 35x - 25 | Factor cubic into linear integer factors if possible. | Find integer roots and factor as (x - r1)(x - r2)(x - r3). | (x - 5)(x - 1)(x - 5) |
Solve for x: 3x^2 + 9x + 0 = 0 | Quadratic with integer roots and non-unit leading coefficient. | Factor or use quadratic formula; roots are integers. | 0, -3 |
Solve for x: 1(x + 14) = 24 | Solve a two-step equation. | Divide both sides by a, then subtract b. | 10 |
Solve for x: sqrt(6x + 160) = 10 | Radical equation; return integer solution if it exists. | Square both sides to get ax + b = c^2, then solve. | -10 |
Simplify 18/6 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | 3 |
What is 221 ÷ -17? | Divide evenly (negative allowed). | Divide 221 by -17. | -13 |
Solve for x: (-9/9)x + -4 = 68 | Linear equation with fractional coefficient. | Subtract c then multiply by b/a (watch sign). | -72 |
Solve the system:
5x + 3y = -8
-2x + -2y = -4 | Find integer x and y that satisfy both equations. | Use substitution or elimination. | x=-7, y=9 |
Solve for x: 1x^3 + 6x^2 + -1x + -30 = 0 | Cubic polynomial factorable to (x - r1)(x - r2)(x - r3). | Find integer root by rational root theorem then factor. | 2, -3, -5 |
Solve for x: 1x^3 + -4x^2 + -11x + 30 = 0 | Cubic polynomial factorable to (x - r1)(x - r2)(x - r3). | Find integer root by rational root theorem then factor. | 2, -3, 5 |
What is -319 ÷ -29? | Divide evenly (negative allowed). | Divide -319 by -29. | 11 |
Find smallest non-negative x such that 20x ≡ 8 (mod 24) | Solve linear congruence. | Use gcd and modular inverse on reduced modulus (if solution exists). | 4 |
What is 154 ÷ -22? | Divide evenly. | Divide 154 by -22. | -7 |
What is -238 ÷ 14? | Divide evenly. | Divide -238 by 14. | -17 |
Solve for x: -7x + 12 = 131 | Solve the linear equation. | Subtract b from both sides, then divide by a. | -17 |
Solve the system:
-5x + 2y = 5
1x + -3y = 25 | Find integer x and y that satisfy both equations. | Use substitution or elimination. | x=-5, y=-10 |
Find smallest non-negative x such that -15x ≡ 0 (mod 9) | Solve linear congruence. | Use gcd and modular inverse on reduced modulus (if solution exists). | 0 |
Find smallest non-negative x such that 1x ≡ 11 (mod 12) | Solve linear congruence. | Use gcd and modular inverse on reduced modulus (if solution exists). | 11 |
Solve for x: (5/12)x + 7 = 37 | Linear equation with fractional coefficient. | Subtract c then multiply by b/a (watch sign). | 72 |
What is 57 ÷ -19? | Divide evenly (negative allowed). | Divide 57 by -19. | -3 |
Solve for x: log base 2 of x = 1 | Compute x from the log equation (integer-safe). | x = 2^1. | 2 |
Solve for x: (5x + 62)/(3x + -6) = -1 | Rational equation that reduces to a linear equation. | Cross-multiply to get linear equation and solve for x (ensure denominator ≠ 0). | -7 |
Simplify: 3x + 7x + 3y | Combine like terms. | Add coefficients of like terms (x with x, y with y). | 10x + 3y |
Convert the improper fraction 22/11 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. | 2 |
Solve for x: sqrt(-4x + 40) = 8 | Radical equation; return integer solution if it exists. | Square both sides to get ax + b = c^2, then solve. | -6 |
What is 144% of 25? | Compute the percent value (integer-safe). | Compute 144/100 × 25. | 36 |
What is -3^4? | Evaluate the integer exponent. | Multiply -3 by itself 4 times (0th power = 1). | 81 |
Solve: |-5x + 9| ≤ 11 | Find integer range of x satisfying the absolute inequality. | Isolate and produce interval then take integer bounds. | 0 ≤ x ≤ 4 |
Solve for x: (-6/12)x + -4 = 32 | Linear equation with fractional coefficient. | Subtract c then multiply by b/a (watch sign). | -72 |
Solve for x: 4^x = 16 | Solve the exponential equation for integer x. | Find x such that repeated multiplication of 4 yields 16. | 2 |
Convert the improper fraction 230/11 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 10/11 |
Factor the polynomial: 1x^2 + 6x + 9 | Factor quadratic into integer linear factors. | Find integers r1 and r2 such that polynomial = (x - r1)(x - r2). | (x - -3)(x - -3) |
Solve: 9x + 6 ≤ -93 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x ≤ -11 |
What is -216 ÷ 24? | Divide evenly. | Divide -216 by 24. | -9 |
Solve for x: -6x + -29 = -143 | Solve the linear equation. | Subtract b from both sides, then divide by a. | 19 |
What is -113 mod 10 (remainder after division)? | Compute remainder when a is divided by b. | Divide -113 by 10 and take the integer remainder 0..9. | 7 |
What is -7 × -50? | Multiply the numbers. | Multiply -7 by -50. | 350 |
Solve: 7x + -9 ≤ -114 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x ≤ -15 |
Solve for x: 1x^3 + 0x^2 + -19x + -30 = 0 | Cubic polynomial factorable to (x - r1)(x - r2)(x - r3). | Find integer root by rational root theorem then factor. | -2, 5, -3 |
What is -15 ÷ 15? | Divide evenly (negative allowed). | Divide -15 by 15. | -1 |
Solve: -1x + 14 < 8 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x < 6 |
Solve: -10x + 11 ≤ 61 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x ≤ -5 |
Find one integer solution to 9x + 4y = -35 | Linear Diophantine equation (return one integer pair). | Find any particular solution (here constructed). | x=-3, y=-2 |
Solve for x: log base 2 of x = 2 | Compute x from the log equation (integer-safe). | x = 2^2. | 4 |
Solve for x: -3(x + 12) = 3 | Solve a two-step equation. | Divide both sides by a, then subtract b. | -13 |
What is 114 mod 4 (remainder after division)? | Compute remainder when a is divided by b. | Divide 114 by 4 and take the integer remainder 0..3. | 2 |
Solve: |3x + 12| = 15 | Find integer solutions x, if any. | Set inside equal to ±c and solve both linear equations. | 1, -9 |
What is 90 + 105? | Add the two numbers. | Add 90 and 105. | 195 |
If each item costs 15 dollars and the total cost is 270, how many items were bought? | Solve using algebra. | Divide total cost by price per item. | 18 |
Solve for x: 2x^2 - 22x + 60 = 0 | Solve the quadratic equation with integer roots. | Factor or use quadratic formula (should factor cleanly). | 5, 6 |
Solve for x: 1x^3 + -6x^2 + 5x + 12 = 0 | Cubic polynomial factorable to (x - r1)(x - r2)(x - r3). | Find integer root by rational root theorem then factor. | 4, -1, 3 |
Solve for x: 3x^2 + 15x + -18 = 0 | Quadratic with integer roots and non-unit leading coefficient. | Factor or use quadratic formula; roots are integers. | 1, -6 |
Solve for x: (8/4)x + 10 = -38 | Linear equation with fractional coefficient. | Subtract c then multiply by b/a (watch sign). | -24 |
Solve for x: log base 3 of x = 6 | Compute x from the log equation (integer-safe). | x = 3^6. | 729 |
What is 143 - -197? | Subtract the second number from the first. | Subtract -197 from 143. | 340 |
Convert the improper fraction 39/7 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. | 5 4/7 |
Solve: |-1x + -16| ≤ 16 | Find integer range of x satisfying the absolute inequality. | Isolate and produce interval then take integer bounds. | -32 ≤ x ≤ 0 |
What is -1^4? | Evaluate the integer exponent. | Multiply -1 by itself 4 times (0th power = 1). | 1 |
What is 101% of 200? | Compute the percent value (integer-safe). | Compute 101/100 × 200. | 202 |
Simplify 45/9 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | 5 |
What is -51 - -174? | Subtract the second number from the first. | Subtract -174 from -51. | 123 |
Simplify 30/5 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | 6 |
Solve for x: 2x^2 + 10x - 28 = 0 | Solve the quadratic equation with integer roots. | Factor or use quadratic formula (should factor cleanly). | 2, -7 |
Solve for x: 2^x = 1 | Solve the exponential equation for integer x. | Find x such that repeated multiplication of 2 yields 1. | 0 |
What is -221 ÷ 13? | Divide evenly. | Divide -221 by 13. | -17 |
Simplify -55/5 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | -11 |
Find smallest non-negative x such that -17x ≡ 1 (mod 12) | Solve linear congruence. | Use gcd and modular inverse on reduced modulus (if solution exists). | 7 |
What is 0 ÷ -10? | Divide evenly (negative allowed). | Divide 0 by -10. | 0 |
Simplify 15/5 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | 3 |
What is 14 - 36? | Subtract the second number from the first. | Subtract 36 from 14. | -22 |
Factor the polynomial: x^3 - 7x^2 + 2x - -40 | Factor cubic into linear integer factors if possible. | Find integer roots and factor as (x - r1)(x - r2)(x - r3). | (x - 5)(x - 4)(x - -2) |
Solve for x: -4(x + 15) = -44 | Solve a two-step equation. | Divide both sides by a, then subtract b. | -4 |
Solve: |-4x + -4| ≤ 13 | Find integer range of x satisfying the absolute inequality. | Isolate and produce interval then take integer bounds. | -4 ≤ x ≤ 2 |
If each item costs 7 dollars and the total cost is -77, how many items were bought? | Solve using algebra. | Divide total cost by price per item. | -11 |
Solve for x: log base 3 of x = 3 | Compute x from the log equation (integer-safe). | x = 3^3. | 27 |
Convert the improper fraction 9/1 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. | 9 |
Solve the system:
-3x + 3y = 21
2x + 1y = -23 | Find integer x and y that satisfy both equations. | Use substitution or elimination. | x=-10, y=-3 |
Solve for x: 1x^3 + 11x^2 + 39x + 45 = 0 | Cubic polynomial factorable to (x - r1)(x - r2)(x - r3). | Find integer root by rational root theorem then factor. | -3, -5, -3 |
Solve: 8x + 16 > -32 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x > -6 |
Solve for x: (5x + -20)/(-7x + -4) = 0 | Rational equation that reduces to a linear equation. | Cross-multiply to get linear equation and solve for x (ensure denominator ≠ 0). | 4 |
Solve: |-1x + 6| ≤ 5 | Find integer range of x satisfying the absolute inequality. | Isolate and produce interval then take integer bounds. | 1 ≤ x ≤ 11 |
Solve for x: -4x^2 + -40x + -96 = 0 | Quadratic with integer roots and non-unit leading coefficient. | Factor or use quadratic formula; roots are integers. | -6, -4 |
If each item costs 7 dollars and the total cost is -42, how many items were bought? | Solve using algebra. | Divide total cost by price per item. | -6 |
Solve for x: -3x^2 + -9x + 0 = 0 | Quadratic with integer roots and non-unit leading coefficient. | Factor or use quadratic formula; roots are integers. | 0, -3 |
Convert the improper fraction 4/2 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. | 2 |
Solve for x: 1x + 29 = 40 | Solve the linear equation. | Subtract b from both sides, then divide by a. | 11 |
What is 0 ÷ 27? | Divide evenly. | Divide 0 by 27. | 0 |
Solve: |-6x + -1| = 13 | Find integer solutions x, if any. | Set inside equal to ±c and solve both linear equations. | 2 |
Solve for x: 2^x = 16 | Solve the exponential equation for integer x. | Find x such that repeated multiplication of 2 yields 16. | 4 |
What is 57 × -7? | Multiply the numbers. | Multiply 57 by -7. | -399 |
Solve for x: -2x^2 + 30x - 112 = 0 | Solve the quadratic equation with integer roots. | Factor or use quadratic formula (should factor cleanly). | 7, 8 |
What is 2^3? | Evaluate the integer exponent. | Multiply 2 by itself 3 times (0th power = 1). | 8 |
Solve for x: 4x + 11 = 15 | Solve the linear equation. | Subtract b from both sides, then divide by a. | 1 |
Convert the improper fraction 29/2 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. | 14 1/2 |
Solve for x: 8(x + 7) = 176 | Solve a two-step equation. | Divide both sides by a, then subtract b. | 15 |
Find smallest non-negative x such that 18x ≡ 0 (mod 6) | Solve linear congruence. | Use gcd and modular inverse on reduced modulus (if solution exists). | 0 |
What is 6^5? | Evaluate the integer exponent. | Multiply 6 by itself 5 times (0th power = 1). | 7776 |
If each item costs -3 dollars and the total cost is -90, how many items were bought? | Solve using algebra. | Divide total cost by price per item. | 30 |
Simplify -18/2 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | -9 |
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