Final answer:
To find the average of integers a, b, and c, we can use the information given in the problem. By solving a system of equations, we can determine the values of a, b, and c that satisfy the given conditions. There are infinitely many solutions for a, b, and c, as long as c is equal to 4.5 + b.
Step-by-step explanation:
To find the average of integers a, b, and c, we need to consider the given conditions. The average of a and 2b is 7, which can be written as (a+2b)/2 = 7. Similarly, the average of a and 2c is 8, which can be written as (a+2c)/2 = 8. To solve these equations, we can use the method of substitution. From the first equation, we can express a in terms of b as a = 7 - 2b, and substitute this value in the second equation. This gives (7 - 2b + 2c)/2 = 8. Simplifying this equation, we get 7 - 2b + 2c = 16. Rearranging the terms, we have -2b + 2c = 16 - 7, which simplifies to -2b + 2c = 9.
Now, we can use this equation along with the expression for a in terms of b to find the values of a, b, and c. Substituting a = 7 - 2b and -2b + 2c = 9 into the equation -2b + 2c = 9, we can solve for b and c. Rearranging the equation, we have 2c = 9 + 2b. Dividing both sides by 2, we get c = (9 + 2b)/2, which simplifies to c = 4.5 + b.
Substituting c = 4.5 + b into the expression for a in terms of b, we can solve for a. Substituting c = 4.5 + b and a = 7 - 2b into the equation a = 7 - 2b, we get 7 - 2b = 7 - 2b. This equation is true for any value of b. Therefore, there are infinitely many solutions for a, b, and c, as long as c is equal to 4.5 + b.