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The total number of​ restaurant-purchased meals that the average person will eat in a​ restaurant, in a​ car, or at home in a year is 191
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The total number of these meals eaten in a car or at home exceeds the number eaten in a restaurant by 13 thirty more​ restaurant-purchased meals will be eaten in a restaurant than at home. Find the number of​ restaurant-purchased meals eaten in a​ restaurant, the number eaten in a​ car, and the number eaten at home.

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sqdancefan

Answer:

  • restaurant: 89
  • car: 43
  • home: 59

Step-by-step explanation:

Let r, c, h represent the numbers of purchased meals eaten in a restaurant, car, or home, respectively. The given data tells us ...

  r + c + h = 191

  -r + c + h = 13

  r + 0c - h = 30

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Subtracting the second equation from the first, we have ...

  2r = 178

  r = 89

Substituting into the third equation gives ...

  89 -h = 30

  h = 59

Then substituting into the second equation, we have ...

  -89 +c +59 = 13

  c = 43

The number of purchased meals eaten in a restaurant is 89; in a car, 43; and at home, 59.

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Comment on the solution

We have used an ad hoc method of solution of these equations. This is a result of the observation that the only difference in the first two equations is the sign of "r". Knowing r, the last equation lets us find h. Then we can use either of the first two equations to find c.

There are algorithmic methods of solving a system of 3 equations in 3 unknowns. One of these is Cramer's Rule. Row reduction or elimination techniques are not so different from what we have done. The difference is in the presentation. For some of the elimination techniques, a matrix form, or augmented matrix form, of the equations is used.

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