James has $78. He nees to buy some cds and tapes. He can buy 4 cds and 3 tapes and has $3 left or buy 3 cds and 5 tapes and has $13 left. Using system equation, find the cost of each.
To solve this problem using a system of equations, we need to assign variables to the unknowns and set up two equations based on the given information.
Let's assume the cost of a CD is represented by "x" dollars, and the cost of a tape is represented by "y" dollars.
From the information given, we can set up the following equations:
Equation 1: 4x + 3y = 75 ($78 - $3 left after buying 4 CDs and 3 tapes)
Equation 2: 3x + 5y = 65 ($78 - $13 left after buying 3 CDs and 5 tapes)
Now, we have a system of equations:
4x + 3y = 75
3x + 5y = 65
To solve this system, we can use the method of substitution or elimination.
Let's solve it using the method of elimination:
Multiply Equation 1 by 3 and Equation 2 by 4 to make the coefficients of the "x" terms the same:
12x + 9y = 225
12x + 20y = 260
Now, subtract Equation 1 from Equation 2:
(12x + 20y) - (12x + 9y) = 260 - 225
11y = 35
Divide both sides by 11:
y = 35/11
y = 3.18 (rounded to two decimal places)
Now, substitute the value of y into Equation 1:
4x + 3(3.18) = 75
4x + 9.54 = 75
4x = 75 - 9.54
4x = 65.46
x = 65.46/4
x = 16.37 (rounded to two decimal places)
Therefore, the cost of each CD is approximately $16.37, and the cost of each tape is approximately $3.18.