At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
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.
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
