Emilia and Jessica are hosting events that are catered by the same company. Emilia plans to have 96 adults and 42 children attend, so the total projected cost of her meals is $3,336. Jessica has 79 adults and 74 children on her guest list, so she will pay the caterer $3,534. How much does the caterer charge for 1 adult meal and for 1 child's meal?
Let the cost of 1 adult meal be A and the cost of 1 child's meal be C.
From the first statement, we know that 96A + 42C = 3,336.
From the second statement, we know that 79A + 74C = 3,534.
To prepare these equations we've carried out the following step: "the total projected cost of her meals is $3,336" ==> 96A + 42C = 3,336 and "she will pay the caterer $3,534" ==> 79A + 74C = 3,534.
Multiplying the first equation by 2, we have 192A + 84C = 6,672.
Multiplying the second equation by 3, we have 237A + 222C = 10,602.
Multiplying the first equation by 79 and the second equation by 96 to simplify, we get 15168A + 6588C = 524,544 and 22656A + 22092C = 1,017,792.
If we subtract 15168A + 6588C = 524,544 from 22656A + 22092C = 1,017,792, we obtain (22656 - 15168)A + (22092 - 6588)C = (1,017,792 - 524,544), or 7472A + 15504C = 493,248.
Multiplying both sides of 7472A + 15504C = 493,248 by 3, we get 22416A + 46512C = 1,479,744.
If we subtract 237A + 222C = 10,602 from 22416A + 46512C = 1,479,744, we obtain (22416 - 237)A + (46512 - 222)C = (1,479,744 - 10,602), or 22179A + 46290C = 1,469,142.
Now we take the second equation (22179A + 46290C = 1,469,142) multiplied by -108 and add that to the equation 15168A + 6588C = 524,544. This gives us (-108 * 22179A) + (-108 * 46290C) = (-108 * 1,469,142) and 15168A + 6588C = 524,544.
Using the distributive property simplifies the first equation to -2,391,732A - 4,997,520C = -158,718,936, while the second equation remains as 15168A + 6588C = 524544.
Adding the two equations, we get (-2,391,732A - 4,997,520C) + (15168A + 6588C) = (-158,718,936 + 524,544), or -2,376,564A - 4,990,932C = -158,194,392.
Dividing both sides by -2,376,564 gives us A + (4,990,932 / -2,376,564)C = (-158,194,392 / -2,376,564) or A - 2.1C = 66.6464.
This simplifies to A = 2.1C + 66.6464.
Substituting A = 2.1C + 66.6464 into the equation 79A + 74C = 3,534 gives us 79(2.1C + 66.6464) + 74C = 3,534.
Expanding the left side of the equation, we get 165.9C + 5,263.9136 + 74C = 3,534.
Combining like terms gives us 239.9C + 5263.9136 = 3534.93.
Simplifying further, we have 239.9C - 3534.93 + 5263.9136 = 0.
Combining more like terms yields 239.9C + 1728.9836 = 0.
Subtracting 1728.9836 from both sides of the equation, we have 239.9C = -1,728.9836.
Finally, dividing both sides by 239.9 gives us C = -1,728.9836 / 239.9, or C = -7.204.
Therefore, the caterer charges $7.204 for 1 child's meal.
Substituting C = -7.204 into A = 2.1C + 66.6464 gives us A = 2.1(-7.204) + 66.6464.
Expanding the right side of the equation, we get A = -15.1284 + 66.6464.
Combining like terms yields A = 51.518.
Therefore, the caterer charges $51.518 for 1 adult meal. Answer: \boxed{7.204, 51.518}.