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Sagot :
Let's examine the given system of linear equations step-by-step to understand what they represent:
[tex]\[ \begin{aligned} 5x + 3y & = 210 \qquad \text{(Equation 1)} \\ x + y & = 60 \qquad \text{(Equation 2)} \end{aligned} \][/tex]
Interpreting the Equations:
1. Equation 1: [tex]\(5x + 3y = 210\)[/tex]
- This equation implies that we are dealing with two different items (which we will call [tex]\(x\)[/tex] and [tex]\(y\)[/tex]).
- One item (let's call it [tex]\(x\)[/tex]) contributes 5 units towards a total sum.
- The other item (let's call it [tex]\(y\)[/tex]) contributes 3 units towards the same total sum, which amounts to 210.
2. Equation 2: [tex]\(x + y = 60\)[/tex]
- This indicates the total number of those two items is 60.
We need to solve this system to determine [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
Solve for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
From Equation 2:
[tex]\[ y = 60 - x \][/tex]
Substitute [tex]\(y\)[/tex] in Equation 1:
[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]
Solve for [tex]\(y\)[/tex]:
[tex]\[ y = 60 - x \][/tex]
[tex]\[ y = 60 - 15 \][/tex]
[tex]\[ y = 45 \][/tex]
So, [tex]\(x = 15\)[/tex] and [tex]\(y = 45\)[/tex].
Interpreting the Results:
Let's analyze each option to see which matches our equations and solution:
Option A: Involves strings for guitar and banjo.
- Not plausible because the equations specifically reflect a purchase scenario, not instrument strings.
Option B: Involves the sale of chocolates and caramels.
- Incorrect because the system doesn't account for generating money but rather count of items.
Option C: Involves ticket pricing.
- Incorrect interpretation as this is about a simple count of tickets which would be different from a sum operation used.
Option D: Involves paintbrushes bought in packs of 5 and packs of 3.
- Matches our interpretation, as follows:
- The art teacher buys packs of 5 paintbrushes ([tex]\(x\)[/tex]) and packs of 3 paintbrushes ([tex]\(y\)[/tex]).
- She bought a total of [tex]\(60\)[/tex] packs.
- The total number of paintbrushes is [tex]\(210\)[/tex].
Thus, the correct answer is:
D. 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.
[tex]\[ \begin{aligned} 5x + 3y & = 210 \qquad \text{(Equation 1)} \\ x + y & = 60 \qquad \text{(Equation 2)} \end{aligned} \][/tex]
Interpreting the Equations:
1. Equation 1: [tex]\(5x + 3y = 210\)[/tex]
- This equation implies that we are dealing with two different items (which we will call [tex]\(x\)[/tex] and [tex]\(y\)[/tex]).
- One item (let's call it [tex]\(x\)[/tex]) contributes 5 units towards a total sum.
- The other item (let's call it [tex]\(y\)[/tex]) contributes 3 units towards the same total sum, which amounts to 210.
2. Equation 2: [tex]\(x + y = 60\)[/tex]
- This indicates the total number of those two items is 60.
We need to solve this system to determine [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
Solve for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
From Equation 2:
[tex]\[ y = 60 - x \][/tex]
Substitute [tex]\(y\)[/tex] in Equation 1:
[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]
Solve for [tex]\(y\)[/tex]:
[tex]\[ y = 60 - x \][/tex]
[tex]\[ y = 60 - 15 \][/tex]
[tex]\[ y = 45 \][/tex]
So, [tex]\(x = 15\)[/tex] and [tex]\(y = 45\)[/tex].
Interpreting the Results:
Let's analyze each option to see which matches our equations and solution:
Option A: Involves strings for guitar and banjo.
- Not plausible because the equations specifically reflect a purchase scenario, not instrument strings.
Option B: Involves the sale of chocolates and caramels.
- Incorrect because the system doesn't account for generating money but rather count of items.
Option C: Involves ticket pricing.
- Incorrect interpretation as this is about a simple count of tickets which would be different from a sum operation used.
Option D: Involves paintbrushes bought in packs of 5 and packs of 3.
- Matches our interpretation, as follows:
- The art teacher buys packs of 5 paintbrushes ([tex]\(x\)[/tex]) and packs of 3 paintbrushes ([tex]\(y\)[/tex]).
- She bought a total of [tex]\(60\)[/tex] packs.
- The total number of paintbrushes is [tex]\(210\)[/tex].
Thus, the correct answer is:
D. 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.
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