Question

For what value(s) of b can 13 be written as a linear combination of 1, 2b, and 5?

a) Clearly define the problem of expressing 13 as a linear combination of the given terms.
b) Instruct the respondent to determine the possible values of b that satisfy this condition.
c) Encourage a step-by-step explanation or solution process, demonstrating an understanding of linear combinations.

Ensure clarity in the question to prompt a thoughtful exploration and identification of the values of b that make 13 a linear combination of 1, 2b, and 5.

Answer 1

To express 13 as a linear combination of 1, 2b, and 5, we need to find values of b that satisfy the equation 1x + (2b)y + 5z = 13. The possible values of b are any integers that divide evenly into 13 - x - 5z.

Step-by-step explanation:

To express 13 as a linear combination of 1, 2b, and 5, we need to find values of b that satisfy the equation 1x + (2b)y + 5z = 13, where x, y, and z are integers.

Let's start by solving the equation for x, y, and z.

1x + (2b)y + 5z = 13

We can rearrange the equation to isolate y:

(2b)y = 13 - x - 5z

Dividing both sides of the equation by 2b, we get:

y = (13 - x - 5z) / 2b

In order for y to be an integer, 13 - x - 5z must be divisible by 2b.

This means that the possible values of b are any integers that divide evenly into 13 - x - 5z.