MIT

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question:Determine the ordered pair of integers (x, y) that satisfies the system of equations 12x + 21y = 15 and 21x + 12y = 51.

answer:We can solve this system of equations by multiplying the first equation by 21 and the second equation by 12, and then subtracting the second equation from the first to eliminate y: begin{align*} 21(12x + 21y) &= 21(15) 12(21x + 12y) &= 12(51) 252x + 441y &= 315 252x + 144y &= 612 (252x + 441y) - (252x + 144y) &= 315 - 612 297y &= -297 y &= -1 end{align*} Substituting this value of y into the first equation, we can find the value of x: begin{align*} 12x + 21(-1) &= 15 12x - 21 &= 15 12x &= 36 x &= 3 end{align*} Therefore, the ordered pair (x, y) that satisfies the system of equations is boxed{(3, -1)}.The answer is: (3,-1)

question:A vampire drains three people a week. His best friend is a werewolf who eats five people a week, but only fresh ones, never drained ones. How many weeks will a village of x people last them both? If we know the answer to the above question is 9, what is the value of unknown variable x?

answer:The vampire drains three people a week, and the werewolf eats five people a week. The total number of people they consume each week is the sum of the number of people drained by the vampire and the number of people eaten by the werewolf: 3 + 5 = 8. If a village has x people, then the number of weeks it will last them both is x / 8. We are given that the number of weeks is 9, so we can write: x / 8 = 9. Multiplying both sides by 8, we get: x = 72. The value of x is 72. 72 The answer is: 72

question:is Roger’s turn to provide a snack for the baseball team after the game and he has decided to bring trail mix. The trail mix comes in packs of 6 individual pouches. Roger has 13 members on his baseball team, plus x coaches and 2 helpers. How many packs of trail mix does he need to buy? If we know the answer to the above question is 3, what is the value of unknown variable x?

answer:Roger has 13 members on his baseball team, plus x coaches and 2 helpers, so the total number of people who need a snack is 13 + x + 2 = 15 + x. Each pack of trail mix has 6 individual pouches. To find out how many packs of trail mix Roger needs to buy, we can divide the total number of pouches needed by the number of pouches in each pack: (15 + x) / 6. We are given that Roger needs 3 packs of trail mix, so we can write: (15 + x) / 6 = 3. Multiplying both sides by 6, we get: 15 + x = 18. Subtracting 15 from both sides, we get: x = 3. The value of x is 3. 3 The answer is: 3

question:Express 1.overline{03} as a reduced fraction, given that 0.overline{01} is frac{1}{99}.

answer:We can rewrite 1.overline{03} as 1 + 0.overline{03}. Since 0.overline{03} is the repeating decimal equivalent of frac{3}{99}, we have 1 + 0.overline{03} = 1 + frac{3}{99} = frac{99}{99} + frac{3}{99} = frac{102}{99}. To reduce this fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 3. So, we have frac{102}{99} = frac{34}{33}. Therefore, 1.overline{03} is equal to boxed{frac{34}{33}}.The answer is: frac{34}{33}