Re: An auditorium has only orchestra and balcony seating. Tickets for the
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29 Oct 2021, 19:36
Let total number of Orchestra seats = x -> equation A
Let total number of balcony seats be y -> equation B
Given for the first performance of the concert, each orchestra ticket is sold for $80. By this we can calculate the total amount costed for certain number of tickets(which is unknown as of now) as below
$80(x)=The total dollar amount of "x" number of orchesthra tickets. -> equation:1
Now we follow the same above steps for the amount costed for balcony seats.
$50(y)=The total dollar amount costed for "y" number of balcony seats. -> equation:2
Now we are given the total dollar amount costed for both the orchestra and balcony seats combined which is: $49,000.
Now looking at equation 1 and 2, which is the dollar amount of individual seats which are orchestra and balcony, we can recognize that, both amounts combined comes out to be the total dollar amount given in the question. Which is $49,000.
Now let's translate the whole information above into a linear equation. Which will look like below
$80(x) + $50(y) = $49,000 -> equation 3.
Now that we are done with the amount equation, let us take a look at the last part of the question. Which says, "The total number of seats sold sums up to be 800".
Going back to the very start of the explanation, let us take a look at equations A & B. Relating those equation with the information above, we can translate it into linear equation again.
Number of orchestra seats + Number of balcony seats = Given total number of seats. Which will translate to:
x + y = 800 -> equation 4.
Now that we have both the equation 3 & 4 set, let us add both the equations to get what we are asked to find out which is the number of balcony seats. Below is the simplification.
$80(x) + $50(y) = $49,000
x + y = 800
Now let us multiply the entire equation 4 with any of the co-efficients of x and y for easy calculation. Let us multiply the below equation with smaller number which is the number 50. After the multiplication,
$80(x) + $50(y) = $49,000
50(x) + 50(y) = 40,000
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30(x) - 0 = 9000
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Now from the above equation, we solve for "x". Upon solving, we get,
x = 300.
Now, let us substitute the above "x" value in equation 4.
x + y = 800
300 + y = 800
y = 800 - 300 which gives us,
y = 500.
Finally we found out the number of balcony seats out of the total 800.
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