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Sagot :
Let's analyze the system of linear equations and what each variable might represent in the different scenarios:
The system is:
[tex]\[ 5x + 3y = 210 \][/tex]
[tex]\[ x + y = 60 \][/tex]
First, solve the second equation for one of the variables:
[tex]\[ x + y = 60 \implies y = 60 - x \][/tex]
Substitute [tex]\( y = 60 - x \)[/tex] into the first equation:
[tex]\[ 5x + 3(60 - x) = 210 \][/tex]
[tex]\[ 5x + 180 - 3x = 210 \][/tex]
[tex]\[ 2x + 180 = 210 \][/tex]
[tex]\[ 2x = 30 \][/tex]
[tex]\[ x = 15 \][/tex]
Then, substituting [tex]\( x = 15 \)[/tex] back into [tex]\( y = 60 - x \)[/tex]:
[tex]\[ y = 60 - 15 \][/tex]
[tex]\[ y = 45 \][/tex]
So, we have [tex]\( x = 15 \)[/tex] and [tex]\( y = 45 \)[/tex].
Now, let’s interpret these results within each context:
A. An art teacher bought paintbrushes in packs of 5 and packs of 3. She bought a total of 60 packs and now has 210 paintbrushes.
- Let [tex]\( x \)[/tex] be the number of 5-pack paintbrushes and [tex]\( y \)[/tex] be the number of 3-pack paintbrushes.
- According to the equations:
- [tex]\( 5x + 3y = 210 \)[/tex] (total paintbrushes)
- [tex]\( x + y = 60 \)[/tex] (total packs)
- Substituting [tex]\( x = 15 \)[/tex] and [tex]\( y = 45 \)[/tex]:
- [tex]\( 5(15) + 3(45) = 75 + 135 = 210 \)[/tex] (valid)
- [tex]\( 15 + 45 = 60 \)[/tex] (valid)
- This interpretation matches the scenario exactly.
B. A guitar requires 5 strings and a banjo requires 3 strings. An orchestra has a total of 210 strings. One guitar player and one banjo player have 60 strings.
- Number of string instruments wouldn't fit having the total number of strings equation and the musical context.
- [tex]\( x \)[/tex] and [tex]\( y \)[/tex] represent the numbers of guitars and banjos, but the interpretation of total strings does not match properly.
C. A candy store sells boxes of chocolates for [tex]$5 each and boxes of caramels for $[/tex]3 each. In one afternoon, the store sold 210 boxes of candy and made a profit of [tex]$60. - This would mean the total number of boxes sold is 210, which does not match the problem statement. - Equations do not match candy and profit context. D. An audience contains 210 people. Student tickets cost $[/tex]3 each and adult tickets cost $5 each. At one performance, there are 60 more adults than students.
- This implies a solution involving money which isn't equated correctly by people counts.
The correct interpretation is:
A. An art teacher bought paintbrushes in packs of 5 and packs of 3. She bought a total of 60 packs and now has 210 paintbrushes.
This scenario fits perfectly with the solution obtained from the system of equations.
The system is:
[tex]\[ 5x + 3y = 210 \][/tex]
[tex]\[ x + y = 60 \][/tex]
First, solve the second equation for one of the variables:
[tex]\[ x + y = 60 \implies y = 60 - x \][/tex]
Substitute [tex]\( y = 60 - x \)[/tex] into the first equation:
[tex]\[ 5x + 3(60 - x) = 210 \][/tex]
[tex]\[ 5x + 180 - 3x = 210 \][/tex]
[tex]\[ 2x + 180 = 210 \][/tex]
[tex]\[ 2x = 30 \][/tex]
[tex]\[ x = 15 \][/tex]
Then, substituting [tex]\( x = 15 \)[/tex] back into [tex]\( y = 60 - x \)[/tex]:
[tex]\[ y = 60 - 15 \][/tex]
[tex]\[ y = 45 \][/tex]
So, we have [tex]\( x = 15 \)[/tex] and [tex]\( y = 45 \)[/tex].
Now, let’s interpret these results within each context:
A. An art teacher bought paintbrushes in packs of 5 and packs of 3. She bought a total of 60 packs and now has 210 paintbrushes.
- Let [tex]\( x \)[/tex] be the number of 5-pack paintbrushes and [tex]\( y \)[/tex] be the number of 3-pack paintbrushes.
- According to the equations:
- [tex]\( 5x + 3y = 210 \)[/tex] (total paintbrushes)
- [tex]\( x + y = 60 \)[/tex] (total packs)
- Substituting [tex]\( x = 15 \)[/tex] and [tex]\( y = 45 \)[/tex]:
- [tex]\( 5(15) + 3(45) = 75 + 135 = 210 \)[/tex] (valid)
- [tex]\( 15 + 45 = 60 \)[/tex] (valid)
- This interpretation matches the scenario exactly.
B. A guitar requires 5 strings and a banjo requires 3 strings. An orchestra has a total of 210 strings. One guitar player and one banjo player have 60 strings.
- Number of string instruments wouldn't fit having the total number of strings equation and the musical context.
- [tex]\( x \)[/tex] and [tex]\( y \)[/tex] represent the numbers of guitars and banjos, but the interpretation of total strings does not match properly.
C. A candy store sells boxes of chocolates for [tex]$5 each and boxes of caramels for $[/tex]3 each. In one afternoon, the store sold 210 boxes of candy and made a profit of [tex]$60. - This would mean the total number of boxes sold is 210, which does not match the problem statement. - Equations do not match candy and profit context. D. An audience contains 210 people. Student tickets cost $[/tex]3 each and adult tickets cost $5 each. At one performance, there are 60 more adults than students.
- This implies a solution involving money which isn't equated correctly by people counts.
The correct interpretation is:
A. An art teacher bought paintbrushes in packs of 5 and packs of 3. She bought a total of 60 packs and now has 210 paintbrushes.
This scenario fits perfectly with the solution obtained from the system of equations.
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