Alright, let's solve this step-by-step.
1. Understanding the Problem:
- Vlad spent time on two types of homework: history and math.
- He spent 20 minutes on history homework.
- He solved [tex]\( x \)[/tex] math problems.
- Each math problem took 2 minutes to solve.
2. Identifying Variables:
- Let [tex]\( x \)[/tex] represent the number of math problems Vlad solved.
- Let [tex]\( y \)[/tex] represent the total time Vlad spent on his homework.
3. Formulating the Equation:
- Time spent on history homework is a constant 20 minutes.
- Time spent solving math problems is [tex]\( 2 \)[/tex] minutes per problem, so the total time for [tex]\( x \)[/tex] problems is [tex]\( 2x \)[/tex] minutes.
- The total time [tex]\( y \)[/tex] is the sum of the time spent on history and math homework.
- Therefore, the equation to find the total time [tex]\( y \)[/tex] is:
[tex]\[
y = 2x + 20
\][/tex]
4. Identifying Constraints:
- [tex]\( x \)[/tex] represents the number of math problems solved, which can only be non-negative integers (whole numbers) because you can't solve a negative or a fraction of a problem. Thus, [tex]\( x \geq 0 \)[/tex] and [tex]\( x \)[/tex] is an integer.
- [tex]\( y \)[/tex] is the total time spent doing homework. The minimum value for [tex]\( y \)[/tex] occurs when [tex]\( x = 0 \)[/tex], giving [tex]\( y = 20 \)[/tex]. As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] also increases. Therefore, [tex]\( y \geq 20 \)[/tex] and [tex]\( y \)[/tex] is an integer since both components (20 and multiples of 2) are integers.
5. Conclusion:
- The correct equation and constraints are:
[tex]\[
y = 2x + 20
\][/tex]
- [tex]\( x \)[/tex] is any integer greater than or equal to 0.
- [tex]\( y \)[/tex] is an integer greater than or equal to 20.
So, the accurate choice that fits these conditions is:
[tex]\[
y = 2x + 20 ; x \text{ is any integer greater than or equal to 0, and } y \text{ is an integer greater than or equal to 20.}
\][/tex]
