Question

a discount book store sells hardcover books for \\$8 each and paperbacks for \\$4 each. the number of hardcover books, \\(h\\), sold one day was 2 more than four times the number of paperback books, \\(p\\), sold that same day. the total amount of sales for that day was \\$160. jean-paul wrote the following system of linear equations to find the number of each type of book sold.

\\8h + 4p = 160\\
\\p = 4h + 2\\

what is jean-pauls error?

• jean-paul should have written the first equation as \\(h + p = 160\\).
• jean-paul should have written the first equation as \\(4h + 8p = 160\\).
• jean-paul should have written the second equation as \\(h = 4p + 2\\).
• jean-paul should have written the second equation as \\(p + 2 = 4h\\).

Explanation:

Identify variables and given values

Using the Variable Identification knowledge point

• Let \(h\) represent the number of hardcover books sold.
• Let \(p\) represent the number of paperback books sold.
• Hardcover price is \(\$8\) each, and paperback price is \(\$4\) each.
• Total sales amount is \(\$160\).

Translate total sales into an equation

Using the Algebraic Translation knowledge point

$$ 8h + 4p = 160 $$

Translate the relationship between books

Using the System of Equations Translation knowledge point

• The problem states: "The number of hardcover books, \(h\), sold one day was 2 more than four times the number of paperback books, \(p\), sold that same day."
• "Four times the number of paperback books" is \(4p\).
• "2 more than" that amount is \(4p + 2\).
• Therefore, the correct equation is:

$$ h = 4p + 2 $$

Compare with Jean-Paul's system and identify the error

Using the System of Equations Translation knowledge point

• Jean-Paul wrote:

$$
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$$

• His first equation is correct.
• His second equation, \(p = 4h + 2\), incorrectly swaps the variables \(h\) and \(p\).
• He should have written the second equation as \(h = 4p + 2\).

Answer:

• Jean-Paul should have written the first equation as \(h+p=160\).
• Jean-Paul should have written the first equation as \(4h+8p=160\).
• Jean-Paul should have written the second equation as \(h=4p+2\). (Correct answer)
• Jean-Paul should have written the second equation as \(p+2=4h\).