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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineAt a community barbecue, Mrs. Davidson and Mr. Ayala are buying dinner for their families. Mrs. Davidson purchases 22 hot dog meals and 33 hamburger meals, paying a total of $39\$39. Mr. Ayala buys 33 hot dog meals and 33 hamburger meals, spending $45\$45 in all. How much do the meals cost?\newlineHot dog meals cost $\$_____ each, and hamburger meals cost $\$_____ each.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineAt a community barbecue, Mrs. Davidson and Mr. Ayala are buying dinner for their families. Mrs. Davidson purchases 22 hot dog meals and 33 hamburger meals, paying a total of $39\$39. Mr. Ayala buys 33 hot dog meals and 33 hamburger meals, spending $45\$45 in all. How much do the meals cost?\newlineHot dog meals cost $\$_____ each, and hamburger meals cost $\$_____ each.
  1. Define Prices: Let's denote the price of each hot dog meal as hh and the price of each hamburger meal as dd. Mrs. Davidson purchased 22 hot dog meals and 33 hamburger meals for a total of $\$3939. This gives us the equation 2h+3d=392h + 3d = 39.
  2. Mrs. Davidson's Purchase: Mr. Ayala purchased 33 hot dog meals and 33 hamburger meals for a total of $45\$45. This gives us the equation 3h+3d=453h + 3d = 45.
  3. Eliminate Variable: We now have a system of two equations. We need to eliminate one of the variables, hh or dd. We choose to eliminate hh because its coefficients are the same in both equations.
  4. Adjust Equations: To eliminate hh, we multiply the first equation by −32-\frac{3}{2} to match the coefficient of hh in the second equation. This gives us the new equation −3h−4.5d=−58.5-3h - 4.5d = -58.5.
  5. Solve for dd: We now add the new first equation to the second equation to eliminate hh. This gives us −1.5d=−13.5-1.5d = -13.5, or d=9d = 9 after dividing both sides by −1.5-1.5.
  6. Solve for hh: We substitute d=9d = 9 into the first equation and solve for hh. This gives us 2h+27=392h + 27 = 39, or 2h=122h = 12 after subtracting 2727 from both sides, and finally h=6h = 6 after dividing both sides by 22.

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