One week, the music store sold 2 trumpets, 3 clarinets, and 5 violins for $1240. the next week, they sold 3 trumpets, 1 clarinet, and 4 violins for $1027. the following week, they sold 5 trumpets, 7 clarinets, and 2 violins for $2091. find the cost of a trumpet, a clarinet would, and a violin.

One week, the music store sold 2 trumpets, 3 clarinets, and 5 violins for $1240. the next week, they sold 3 trumpets, 1 clarinet, and 4 violins for $1027. the following week, they sold 5 trumpets, 7 clarinets, and 2 violins for $2091.

Let x be the cost of trumpets  

y be the cost of clarinets

z be the cost of violins

Now we frame equations

2x +3y + 5z = 1240 ---> equation 1

3x + 1y + 4z = 1027---> equation 2

5x + 7y + 2z = 2091 ---> equation 3

Now we solve for x,y and z using elimination method

Multiply the second equation by -3 and with first equation

2x +3y + 5z = 1240  

-9x - 3y -12z = -3081

-7x - 7z = -1841  ( divide both sides by -7)

x + y = 263 > equation 4

Multiply the second equation by -7 and add it with third equation

-21x - 7y - 28z = -7189    

 5x + 7y + 2z = 2091  

-16x - 26z = -5098 (divide the whole equation by -2)

8x + 13z = 2549  > equation 5

Now use equation 4  and 5 to eliminate x. Multiply equation 4 by 8

-8x - 8z = -2104

8x + 13z = 2549

5z = 445

z = 89

We know x + z = 263, replace z with 89

x + 89 = 263 ( subtract 89 on both sides)

x = 174

2x +3y + 5z = 1240 ---> equation 1 ( substitute the values of x  and z)

2(174) + 3y + 5(89) = 1240

348 + 3y + 445 = 1240

793 + 3y = 1240 ( subtract 793 on both sides)

3y = 447 ( divide by 3 on both sides)

y = 149

The cost of trumpets  = $174

The cost of clarinets = $149

The cost of violins = $89