Questions from a chapter about \"System of equations\" like substitution and eli
ID: 2896510 • Letter: Q
Question
Questions from a chapter about "System of equations" like substitution and elimination
From a point on a river, two boats are driven in the opposite directions, one at 9 miles per hour and the other at 13 miles per hour. In how many hours will they be 44 miles apart?
A basketball field house seats 15,000. court side seats sell for $8, end zone for $7, and balcony for $4. a full house earns $78000 in ticket revenue. if half the court side and balcony sears and all the end zone seats are sold, the total ticket revenue is $46,000. how many of each type of seats are there?
If you could show me how to do both, that would really help be able to finish this assignment by tonight!
Explanation / Answer
Let the boats be driven for x hours before they are 44 miles apart. The distance travelled by the boat travelling at 9 mph inx hours is 9x miles and the distance travelled by the boat travelling at 13 mph inx hours is 13x miles. Sine the boats are travelling in opposite directions and since the two boats are 44 miles apart, we have 9x + 13x = 44 or 22x = 44. Therefore, x = 2 hours. The answer is 2 hours.
Let the number of courtside seats , end zone seats and bacony seats be x, y and z respectively. Then, we have
x + y + z = 15000 ...( 1 )
8x + 7y + 4z = 78000 ...( 2 )
8 (1/2)x + 7y + 4 ( 1/2)z = 46000 or, 4x + 7y + 2z = 46000 ... (3)
FRom the 1st equation, we have z = 15000 - x - y. On substituting this value of z in the 2nd equation, we get
8x + 7y + 4 ( 15000 - x - y) = 78000 or, 8x + 7y + 60000 - 4x - 4 y = 78000 or, 4x + 3y = 78000 - 60000 or,
4x + 3y = 18000 .... ( 4 ).
On substituting this value of z in the 3rd equation, we get 4x + 7y + 2 ( 15000 - x - y) = 46000 or, 4x + 7y + 30000 - 2x - 2y = 46000 or, 2x + 5y = 46000 - 30000 or, 2x + 5y = 16000 ....( 5)
On multiplying both the sides of equation number 5 by 2 and subtracting the result from equation number 4, we get
4x + 3 y -4x -10y = 18000 - 32000 or, -7y = -14000 or 7y = 14000 or y = 2000. Now from the 5th equation, we get 2x + 5 (2000) = 16000 or, 2x + 10000 = 16000 or 2x = 16000 - 10000 = 6000 or x = 3000. Now we know that z = 15000 - x - y. Therefore z = 15000- 3000 - 2000 = 15000 - 5000 = 10000. Thus, the number of court side seats is 3000, the number of end zone seats is 2000 and the number of balcony seats is 10000. We can verify the result by substituting the values of x, y and z in any of the equations no. 1, 2, or 3 or all of these.
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