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 $y$?
A. $y=2 x+20 ; x$ is any integer greater than or equal to 0, and $y$ is an integer greater than or equal to 20.
B. $y=2 x+20 ; x$ is any real number greater than or equal to 0, and $y$ is any real number greater than or equal to 20.
C. $y=20 x+2 ; x$ is any integer greater than or equal to 0, and $y$ is an integer greater than or equal to 20.
D. $y=20 x+2 ; x$ is any real number greater than or equal to 0, and $y$ is any real number greater than or equal to 20.
Answer (1)
The time spent on math problems is 2 x .
The total time spent on homework is y = 2 x + 20 .
x is a non-negative integer, so x ≥ 0 .
y is an integer greater than or equal to 20, so y ≥ 20 .
The equation is y = 2 x + 20 ; x is any integer greater than or equal to 0 , and y is an integer greater than or equal to 20 .
Explanation
Understanding the Problem Let's break down this problem step by step to understand how to find the equation and constraints for the total time Vlad spent on his homework.
Calculating Time Spent on Math Vlad spent 20 minutes on history, and each math problem took 2 minutes. If he solved x math problems, the time spent on math is 2 × x = 2 x minutes.
Formulating the Equation The total time y is the sum of the time spent on history and math. So, we have the equation: y = 2 x + 20
Determining Constraints on x Now, let's think about the constraints. The number of math problems x must be a non-negative integer because Vlad can't solve a fraction of a problem. So, x can be 0, 1, 2, 3, and so on. x ≥ 0 and x is an integer
Determining Constraints on y Since x is a non-negative integer, the smallest value for x is 0. When x = 0 , y = 2 ( 0 ) + 20 = 20 . As x increases, y also increases. Since x is an integer, 2 x will always be an even number, and y will always be an integer. Therefore, y must be an integer greater than or equal to 20. y ≥ 20 and y is an integer
Final Answer So, the equation is y = 2 x + 20 , where x is any integer greater than or equal to 0, and y is an integer greater than or equal to 20.
Examples
Imagine Vlad is planning his study schedule. He knows he has 20 minutes of history homework and wants to figure out how much time he'll spend in total if he does a certain number of math problems. If each math problem takes 2 minutes, this equation helps him calculate the total time. For example, if Vlad decides to solve 5 math problems, he can calculate the total time as y = 2 × 5 + 20 = 30 minutes. This kind of calculation is useful for time management and planning.
