A value 2 times a smaller number is 2 less than a larger number . Their difference is 13. What is the larger and smaller number?

Question

Stuck on this math problem regarding systems of equations, would really appreciate the help, thanks!

Answer 1

Question: A value 2 times a smaller number is 2 less than a larger number. Their difference in 13. What is the larger and smaller number?

Answer: Let's get some labels in here. Let b be the bigger number, and s be the smaller number.

"A value 2 times a smaller number" should translate to 2s, or 2 times s.

"...is..." implies a statement of equivalence. Let's put an equal sign here.

"2 less than a larger number" should translate to b - 2.

We should translate the first sentence of the problem as 2s = b - 2.

"Their difference is thirteen" suggests that when we subtract the smaller number from the bigger number, we get 13. Translate this as b - s = 13.

We now have a system of equations:

2s = b-2

b - s = 13

Let's rearrange this a bit, to make it easier for the elimination method. Subtract b from both sides in the first equation:

-b + 2s = -2

b - s = 13

This is nicely set up to eliminate b. We get s = 11, and plugging 11 into s in the second equation gets us

b - (11) = 13, or b = 24.

From this, we find that the smaller number is 11, and the bigger number is 24.

Check our work:

"A value 2 times a smaller number is 2 less than a larger number." 2(11) = (24) - 2 ---> 22 =22 is a true statement.

"Their difference in 13." (24) - (11) = 13 ---> 13 = 13 is a true statement. Looking good!

The smaller number is 11, and the bigger number is 24.