Respuesta :
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
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-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
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-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
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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
Using a system of equations, it is found that:
- The cost of a trumpet is of $174.
- The cost of a clarinet is of $149.
- The cost of a violin is of $89.
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For the system of equations, i will say that:
- The cost of a trumpet is x.
- The cost of a clarinet is y.
- The cost of a violin is z.
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2 trumpets, 3 clarinets, and 5 violins for $1240, thus:
[tex]2x + 3y + 5z = 1240[/tex]
3 trumpets, 1 clarinet, and 4 violins for $1027, thus:
[tex]3x + y + 4z = 1027[/tex]
5 trumpets, 7 clarinets, and 2 violins for $2091, thus:
[tex]5x + 7y + 2z = 2091[/tex]
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From the second equation:
[tex]y = 1027 - 3x - 4z[/tex]
Replacing into the other two:
[tex]2x + 3y + 5z = 1240[/tex]
[tex]2x + 3(1027 - 3x - 4z) + 5z = 1240[/tex]
[tex]-7x - 7z = -1841[/tex]
Simplifying by 7:
[tex]x + z = 263 \rightarrow z = 263 - x[/tex]
Also:
[tex]5x + 7y + 2z = 2091[/tex]
[tex]5x + 7(1027 - 3x - 4z) + 2z = 2091[/tex]
[tex]-16x - 26z = -5098[/tex]
[tex]16x + 26z = 5098[/tex]
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Solving for x:
[tex]16x + 26z = 5098[/tex]
[tex]16x + 26(263 - x) = 5098[/tex]
[tex]10x = 1740[/tex]
[tex]x = \frac{1740}{10}[/tex]
[tex]x = 174[/tex]
The cost of a trumpet is of $174.
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Solving for z:
[tex]z = 263 - x = 263 - 174 = 89[/tex]
The cost of a violin is of $89.
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Solving for y:
[tex]y = 1027 - 3x - 4z = 1027 - 3(174) - 4(89) = 149[/tex]
The cost of a clarinet is of $149.
A similar problem is given at https://brainly.com/question/22826010
