Grades 11-12 Video Solutions 2023-2023_11-12

Video Transcription

6. How many pairs of positive integers x and y satisfy the equation x plus 2y equals 2 to the power of 10? For this problem, we know that x equals 2 to the power of 10 minus 2y  because we can take 2y from the left side and move it to the right side. Since we are given that both x and y are positive integers, or more specifically that x is a positive  integer, we know that 2 to the 10 minus 2y, which is the right hand side of the above equation, is greater than 0 because x is a positive integer. Which means that y is less  than 2 to the 9 because we can isolate for y in the previous inequality. Thus, since y is an integer, the maximum value of y is then 2 to the 9 minus 1. What is the minimum  value of y? Well, since y is also a positive integer, the minimum value of y is 1, the minimum positive integer. There is exactly 1 value of x for each value of y in this range.  So that means for all the values of y from 1 to 2 to the 9 minus 1, we'll get a corresponding value of x, and that means our answer is just a number of integers from 1 to 2 to the  9 minus 1, which is 2 to the 9 minus 1, or a.

Video Summary

6. How many pairs of positive integers x and y satisfy the equation x plus 2y equals 2 to the power of 10? For this problem, we know that x equals 2 to the power of 10 minus 2y because we can take 2y from the left side and move it to the right side. Since we are given that both x and y are positive integers, or more specifically that x is a positive integer, we know that 2 to the 10 minus 2y, which is the right hand side of the above equation, is greater than 0 because x is a positive integer. Which means that y is less than 2 to the 9 because we can isolate for y in the previous inequality. Thus, since y is an integer, the maximum value of y is then 2 to the 9 minus 1. What is the minimum value of y? Well, since y is also a positive integer, the minimum value of y is 1, the minimum positive integer. There is exactly 1 value of x for each value of y in this range. So that means for all the values of y from 1 to 2 to the 9 minus 1, we'll get a corresponding value of x, and that means our answer is just a number of integers from 1 to 2 to the 9 minus 1, which is 2 to the 9 minus 1, or a.