What are the two integers?

Question

The larger of two integers is 19 greater than twice the smaller integer. The smaller integer is 5 less than twice the larger integer. What are the two integers?

Answer 1

We will need to call each integer something. Let's let the Larger integer by L, and the Smaller integer be S.

Now we turn the English words into "math".

"The larger of two integers" for us that means "L"

"is 19 greater than" for us that means "L = 19 + "

"twice the smaller integer" is 2S, so our expression becomes L = 19 + 2S

The smaller integer" for us is "S"

"is 5 less than" than means for us, S = _____ - 5

"twice the larger integer" means 2L, and that is what goes on the blank line.

So S = 2L - 5

Now you have two equations and two unknowns:

L = 19 + 2S

S = 2L - 5

If we substitute our expression for "S" in the second equation INTO THE TOP EQUATION, replace the "S" in 2S on top, we will get an expression in only "L"s

L = 19 + 2 (2L - 5)

Distribute the 2 into the parenthesis.

L = 19 + 4L - 10

Combine like terms on the right.

L = 9 + 4L

Put all the "L"s on the left (subtract 4L from both sides)

-3L = 9

Now divide both sides by NEG 3

L = -3

So S = 2L - 5 or 2 (-3) - 5 = -6 - 5 = -11

Final answer: Larger integer is -3, Smaller integer is -11

Check to see if we are correct: Does -11 = 2 (-3) - 5 ? YES

Does -3 = 19 + 2(-11) YES

So we found our two integers.