two records and three tapes cost $31.three records and two tapes cost $29 .find the cost of each record and each tape
2r + 3t = 31
3r + 2t = 29
1st times 2 ---> 4r + 6t = 62
2nd times 3 --> 9r + 6t = 87
subtract them
5r = 25
r = 5
back into the first ---> 10 + 3t = 31
3t = 21
t = 7
record costs $5 ,and a tape costs $7
To solve this problem, we can assign variables to represent the cost of a record and the cost of a tape.
Let's assume that:
- The cost of one record is "R" dollars, and
- The cost of one tape is "T" dollars.
Now we can use these variables to set up a system of equations based on the given information.
From the first piece of information, we know that two records and three tapes cost $31. Using the variables, we can write this equation as:
2R + 3T = 31. Equation 1.
From the second piece of information, we know that three records and two tapes cost $29. Using the variables, we can write this equation as:
3R + 2T = 29. Equation 2.
Now we have two equations with two unknowns (R and T). We can solve this system of equations to find the values of R and T.
To do this, we can use the method of substitution or elimination. Let's use the elimination method:
Multiply both sides of Equation 1 by 2, and multiply both sides of Equation 2 by 3, to make the coefficients of R in both equations the same:
4R + 6T = 62. Equation 3.
9R + 6T = 87. Equation 4.
Now, subtract Equation 3 from Equation 4 to eliminate the T terms:
(9R + 6T) - (4R + 6T) = 87 - 62,
9R + 6T - 4R - 6T = 25,
5R = 25,
R = 25/5,
R = 5.
Now that we have found the value of R, we can substitute it back into one of the original equations (Equation 1 or Equation 2) to find the value of T.
Let's use Equation 1:
2R + 3T = 31,
2(5) + 3T = 31,
10 + 3T = 31,
3T = 31 - 10,
3T = 21,
T = 21/3,
T = 7.
Therefore, the cost of each record is $5, and the cost of each tape is $7.