Question

vlad spent 20 minutes on his history homework and then completely solved x math problems that each took 2 minutes to complete. what is the equation that can be used to find the value of y, the total time that vlad spent on his homework, and what are the constraints on the values of x and y?
\\(\circ\\) \\(y = 2x + 20\\); \\(x\\) is any integer greater than or equal to 0, and \\(y\\) is an integer greater than or equal to 20.
\\(\circ\\) \\(y = 2x + 20\\); \\(x\\) is any real number greater than or equal to 0, and \\(y\\) is any real number greater than or equal to 20.
\\(\circ\\) \\(y = 20x + 2\\); \\(x\\) is any integer greater than or equal to 0, and \\(y\\) is an integer greater than or equal to 20.
\\(\circ\\) \\(y = 20x + 2\\); \\(x\\) is any real number greater than or equal to 0, and \\(y\\) is any real number greater than or equal to 20.

Explanation:

Step1: Analyze the time spent

Vlad spent 20 minutes on history homework and 2 minutes per math problem. Let \( x \) be the number of math problems (must be a non - negative integer, \( x\geq0 \) and \( x\in\mathbb{Z} \)) and \( y \) be the total time. The total time \( y \) is the sum of the 20 minutes for history and \( 2x \) minutes for math problems. So the equation is \( y = 2x+20 \).

Step2: Determine constraints on \( x \) and \( y \)

The number of math problems \( x \) can't be negative, and since we are counting the number of problems, \( x \) should be an integer (you can't solve a fraction of a problem). So \( x \) is any integer greater than or equal to 0. The total time \( y \) is calculated from the number of problems (an integer) and a fixed 20 minutes, so \( y \) will be an integer greater than or equal to 20 (when \( x = 0 \), \( y=20 \), and as \( x \) increases, \( y \) increases by 2 each time).

Answer:

The correct option is: \( y = 2x + 20 \); \( x \) is any integer greater than or equal to 0, and \( y \) is an integer greater than or equal to 20 (corresponding to the first option among the given choices: \( y = 2x+20 \); \( x \) is any integer greater than or equal to 0, and \( y \) is an integer greater than or equal to 20)