Instrument question

Question

One eel, the music store sold 2 trumpets, 3 clarinets, and 5 violins for $1240.00. The next week, they sold 3 trumpets, 1 clarinet, and 4 violins for $1027.00. The following week, they sold 5 trumpets, 7 clarinets, and 2 violins for $2091.00. Find the cost of a trumpet, clarinet, and violin.

Answer 1

2 trumpets, 3 clarinets, and 5 violins for $1240.00 ::

2t + 3c + 5v = 1240   {equation 1}

 

3 trumpets, 1 clarinet, and 4 violins for $1027.00 ::

3t + c + 4v = 1027    {equation 2}

 

5 trumpets, 7 clarinets, and 2 violins for $2091 ::

5t + 7c + 2v = 2091    {equation 3}

 

We have 3 equations with 3 unknowns.  Let's pare this down to 2

equations with 2 unknowns.  Using first equation 1 and 2 then

equation 2 and 3, let's eliminate the c variable.

 

multiply equation 2 by -3 and add to equation 1:

  2t + 3c + 5v = 1240          equation 1

-9t  - 3c  - 12v = -3081      -3(equation 2)

----------------------------

-7t - 7v = -1841

-7(t + v) = -7(263)

t + v = 263        {equation 1a}

 

Multiply equation 2 by -7 and add to equation 3

-21t - 7c - 28v = -7189    -7(equation 2)

   5t + 7c + 2v = 2091    {equation 3}

-----------------------------

-16t - 26v = -5098

-2(8t + 13v) = -2(2549)

8t + 13v = 2549      {equation 2a}

 

We now have 2 equations with 2 unknowns we can use to solve for t and v

 

t + v = 263            Multiply by -8 and add these equations

8t + 13v = 2549

 

-8t - 8v = -2104

 8t + 13v = 2549

-----------------------

5v = 445

v = $89 per violin

 

t + v = 263

t + 89 = 263

t = $174 per trumpet

 

Using one of the original equations and the above information solve for c

2t + 3c + 5v = 1240   {equation 1}

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

348 + 3c + 445 = 1240

3c = 1240 - 793

3c = 447

c = $149 per clarinet

Answer 2

Hi! This problem seems like it is a substitution problem. Let x= the price of a trumpet, y= the price of a clarinet, and z= price of a violin. Here are the three equations needed from the problem: 

 

2x+3y+5z=1240

 

3x+y+4z= 1027

 

5x+7y+2z= 2091

 

First, Lets manipulate the first two equations so that a term can cancel out. Y seems like the easiest term to eliminate temporary. We can worry about the third equation later. Add the two equations after making one term cancel. 

 

2x+3y+5z=1240

-3( 3x+y+4z = 1027) 

 

2x+3y+5z=1240

-9x-3y-12z= -3081       (the y cancels out, which is important)

 

-7x-7z= -1841  

 

Now lets save this equation for later and now work on the third equation from the beginning. Now we compare the second and third equation from the start and find out how to cancel y again. 

 

3x+y+4z=1027

5x+7y+2z=2091

 

Our goal is to cancel y. So lets try multiplying the first equation to -7. After that, we can add. 

 

-21x-7y-28z= -7189

5x+7y+2z= 2091

 

-16x-26z=-5098

 

 

Now, we use the other equation that we set aside earlier and compare it with this one and see which other term we can cancel out. 

 

-7x-7z=-1841

-16x-26z=-5098

 

Lets eliminate and cancel out Z. So we multiply the first equation to -26/7 and add. 

 

26x+26z=6838

-16x-26z=-5098

 

We get: 10x= 1740

solve for x:

x= 174, this implies that the cost of a trumpet is $174. Lets use this term to calculate the others. So now we plug in x to the equation -7x-7z=-1841, or you can also use the other one: -16x-26z= -5098. 

 

-7(174) -7z=-1841

z= 89 ; hence the cost of a violin is $89. Now we use both Z and X to get our final term. Use any of the top three equations in which we started with. 

 

2x+3y+5z= 1240

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

 

y= 149, hence clarient costs $149. 

 

We now have all three, it is strongly suggested to check work after. So double check by plugging x,y, and z into the second/third equation from the start. 

 

Final answer:

Cost of Trumpet: $174

Cost of Clarinet: $149

Cost of Violin: $89