Five times the larger number is 11 more than three times the smaller. Also, three times the smaller is 16 less than four times the larger. What are the numbers?

Answer 1

We need a variable for the larger (L) and for the smaller (S).

"Five times the larger" becomes 5L

"is" becomes =

"eleven more than" becomes ____ + 11

"three times the smaller number" becomes 3S

Putting the first sentence all together, it becomes

5L = 3S + 11

"Three times the smaller is" becomes 3S =

"sixteen less than" becomes _______ - 16

"four times the larger" becomes 4L

Putting the second sentence all together, it becomes

3S = 4L - 16

Now we have a system of equtions with two unknowns, L and S. We want all of the variables on one side and all of the constants on the other, so these two equations rearrange into

5L - 3S = 11

-4L + 3S = -16

Adding vertically, the S terms cancel and we are left with

L = -5

Plug that back into 3S = 4L-16 and find 3S = 4(-5) - 16 = -36

3S = -36, so S = -12

Final answer:

Larger number is -5

Smaller number is -12