Question

systems of linear equations: mastery test

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$$\begin{cases} 5x + 3y = 210 \\\\ x + y = 60 \\end{cases}$$

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• 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.
• a candy store sells boxes of chocolates for \\$5 each and boxes of caramels for \\$3 each. in one afternoon, the store sold 210 boxes of candy and made a profit of \\$60.
• an audience contains 210 people. student tickets cost \\$3 each and adult tickets cost \\$5 each. at one performance, there are 60 more adults than students.
• an art teacher bought paintbrushes in packs of 5 and packs of 3.

Explanation:

⚡ Using what you learned: introduction to systems of equations

Step 1: Analyze the system of equations

We are given the system:

$$
LATEXBLOCK0
$$

Here, \(x\) and \(y\) represent two unknown quantities.

• The second equation, \(x + y = 60\), means the sum of these two quantities is \(60\).
• The first equation, \(5x + 3y = 210\), means that when we multiply the first quantity by \(5\) and the second quantity by \(3\), the total is \(210\).

Step 2: Evaluate the first option

• Scenario: 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.
• Translation: Let \(x\) be the number of guitars and \(y\) be the number of banjos.
• Total strings: \(5x + 3y = 210\).
• The statement "One guitar player and one banjo player have \(60\) strings" does not translate to \(x + y = 60\) (which would mean the total number of instruments is \(60\)). This option is incorrect.

Step 3: Evaluate the second option

• Scenario: A candy store sells boxes of chocolates for \(\$5\) each and boxes of caramels for \(\$3\) each. In one afternoon, the store sold \(210\) boxes of candy and made a profit of \(\$60\).
• Translation: Let \(x\) be the number of chocolate boxes and \(y\) be the number of caramel boxes.
• Total boxes sold: \(x + y = 210\).
• Total profit/revenue: \(5x + 3y = 60\).
• This reverses the constants (\(210\) and \(60\)) compared to our system. This option is incorrect.

Step 4: Evaluate the third option

• Scenario: An audience contains \(210\) people. Student tickets cost \(\$3\) each and adult tickets cost \(\$5\) each. At one performance, there are \(60\) more adults than students.
• Translation: Let \(x\) be the number of adults and \(y\) be the number of students.
• Total people: \(x + y = 210\).
• "There are \(60\) more adults than students": \(x = y + 60\).
• This does not match our system. This option is incorrect.

Step 5: Evaluate the fourth option (partially cut off)

• Scenario: An art teacher bought paintbrushes in packs of \(5\) and packs of \(3\).
• Complete context: Let \(x\) be the number of packs of \(5\) paintbrushes and \(y\) be the number of packs of \(3\) paintbrushes.
• Total paintbrushes bought: \(5x + 3y = 210\).
• Total number of packs bought: \(x + y = 60\).
• This perfectly matches both equations in the system:

$$
LATEXBLOCK1
$$

Answer:

The correct option is the fourth one:
An art teacher bought paintbrushes in packs of 5 and packs of 3... (representing a total of 60 packs and 210 paintbrushes).