Question 1192189: The following table shows the amount spent by four U.S. airlines to fly one available seat 1 mile in the second quarter of 2014.† Set up a system and then solve using technology. HINT [See the technology note accompanying Example 1.]
Airline United Continental American JetBlue Southwest
Cost (¢) 14.9 14.6 11.9 12.4
Suppose that, on a 3,000-mile New York to Los Angeles flight, United Continental, American, and Southwest flew a total of 235 empty seats, costing them a total of $99,060. If United Continental had three times as many empty seats as American, how many empty seats did each of these three airlines carry on its flight?
+ United Continental _______ empty seats
+ American ______ empty seats
+ Southwest _______ empty seats
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
**1. Set up the variables:**
* Let *u* represent the number of empty seats for United Continental.
* Let *a* represent the number of empty seats for American.
* Let *s* represent the number of empty seats for Southwest.
**2. Set up the equations:**
* **Equation 1 (Total empty seats):** u + a + s = 235
* **Equation 2 (United Continental empty seats):** u = 3a
* **Equation 3 (Total cost of empty seats):** 0.149 * 3000 * u + 0.146 * 3000 * a + 0.124 * 3000 * s = 99060
**3. Simplify Equation 3:**
447u + 438a + 372s = 99060
**4. Substitute Equation 2 into Equation 1 and Equation 3:**
* **New Equation 1:** 3a + a + s = 235 => 4a + s = 235
* **New Equation 3:** 447(3a) + 438a + 372s = 99060 => 1341a + 438a + 372s = 99060 => 1779a + 372s = 99060
**5. Solve the system of two equations (New Equation 1 and New Equation 3):**
One way to do this is using substitution or elimination. Let's use substitution:
* From New Equation 1: s = 235 - 4a
Substitute this into New Equation 3:
1779a + 372(235 - 4a) = 99060
1779a + 87420 - 1488a = 99060
291a = 11640
a = 40
Now substitute a = 40 back into the equation for s:
s = 235 - 4(40)
s = 235 - 160
s = 75
Finally, substitute a = 40 back into the equation for u:
u = 3(40)
u = 120
**Solution:**
* United Continental: 120 empty seats
* American: 40 empty seats
* Southwest: 75 empty seats
Answer by ikleyn(52810)  (Show Source):
You can put this solution on YOUR website! .
As the problem is solved in the post by @CPhill, it is one of possible ways/methods,
but there is another, much more effective and more impressive way to solve the problem
using only one equation from the very beginning, instead of three.
Also, it is worth to notice one very important fact:
in the problem the information about 4 companies is given, but actually only three
companies participate in the game. The fourth company, Jet Blue, is introduced
and does present only to confuse the reader and to distract attention.
Let x be the number of empty seats for American.
Then the number of empty seats for United Continental is 3x,
and the number of empty seats for Southwest is (235 - x - 3x) = 235 - 4x.
Now write the total cost equation for empty seats
United Cont. American Southwest
3000*0.149*(3x) + 3000*0.146*x + 3000*0.124*(235-4x) = 99060 dollars.
Simplify and find x
1341x + 438x + 87420 - 1488x = 99060
291x = 99060 - 87420 = 11640.
x = 11640/191 = 40.
So, the number of empty seats for American is 40; for United Continental 3*40 = 120,
and for Southwest 235-4*40 = 75.
Solved.
Notice that in this solution the system of three equations is not even mentioned.
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