This question involves a lot of decimals that are atypical for this type of question. I rounded along the way, so answers may vary slightly. In any event, here is the answer in essence. I had to delete my previous ansewr so I could add a new one.
For elimination, we are looking for three different functions using the same set of data. In laymen's terms, we want to separate the three different "things." Here, our "things" are child tickets, adult ticket, and senior tickets. We then have two sets of criteria: number of tickets and price per ticket. We then have three different circumstances: Monday, Tuesday, Wednesday. Let's group these together. The problem tells us child tickets are "x," adult "y," and senior "z."
Our three equations will be based upon the three different days.
Monday: 50x + 100y + 25z = 1150
Tuesday: 30x + 150y + 20z = 1620
Wednesday: 20x + 120y + 30z = 1370.
Then line them up.
50x + 100y + 25z = 1150
30x + 150y + 20z = 1620
20x + 120y + 30z = 1370.
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Take the first two (any order is fine, but it's easier to start with 1 and 2). We need to perform elimination with these two.
50x + 100y + 25z = 1150
30x + 150y + 20z = 1620
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To perform this, we must make one variable (what I like to call) "equal opposites."
This is akin to a common denominator in fraction addition/subtraction. This is easiest learned by example.
50x + 100y + 25z = 1150 <== multiply this entire row by 3
30x + 150y + 20z = 1620 <== multiply this entire row by -5
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In the result, notice the 150x and -150x. These are as I call them "equal opposites."
150x + 300y + 75z = 3450
-150x + -750y - 100z = -8100. Now subtract the bottom row from the top.
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0x - 450y - 25z = -4650.
Eliminate the 0x and we are left with ===> -450y - 25z = -4650, or 450y + 25z = 4650 (I simply multiplied the whole equation by -1). Set this equation to the side for a minute.
Now take equations 2 and 3. Again, it doesn't matter as long as you don't use the same two.
30x + 150y + 20z = 1620
20x + 120y + 30z = 1370.
I need to do the same thing, make "equal opposites," although I have to bear in mind that I must eliminate the "x" variable. If you eliminated "y" or "z" in equations 1 and 2, then you must eliminate "y" or "z" at this step. I will multiply the top row by 2 and the bottom row by -3.
Result: 60x + 300y + 40z = 3240
-60x -360y -90z = -4110
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0x - 60y - 50z = -870.
Final answer: 60y + 50z = 870.
Now we have two equations, 450y + 25z = 4650 and 60y + 50z = 870, both of which have only y and z variables.
Perform the same type of elimination on these two.
450y + 25z = 4650. Multiply by -2 so we can eliminate "z."
660y + 50z = 870.
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-900y - 50z = -9300
60y + 50z = 870
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-840y = -8430. Y = 10.04.
Plug in 10.04 for y in any of the equations we just discovered. I'll use 60y + 50z = 870.
60(10.04) + 50z = 870
602.4 + 50z = 870
50z = 267.6. z = 5.34. [this came out to 5.35 because I rounded along the way, but you neeed 5.34 to make it work].
Now, to find "x," simply plug in 5.35 for "z" and 10.04 for "x" in any of the original equations.
20x + 120y + 30z = 1370.
20x + 120(10.04) + 30(5.34) = 1370 20x + 1365 = 1370
20x + 1365 = 1370
20x = 5
x = 0.25
Answer: x = 0.25; y = 10.04; z = 5.35
