# The sum of two numbers is 12. When three times the first number is added to 5 times the second number, the resultant number is 44. How do you find the two numbers?

Mar 22, 2018

The first number is $8$ and the second number is $4$

#### Explanation:

We will turn the word problem into an equation to make it easier to solve. I am going to abbreviate "first number" to $F$ and "second number to $S$.

$\stackrel{F + S}{\overbrace{\text{the sum of the two numbers" stackrel(=)overbrace"is"stackrel(12)overbrace"12}}}$

AND:

$\stackrel{3 F}{\overbrace{\text{three times the first number" " " stackrel(+) overbrace"is added to" " "stackrel(5S)overbrace"five times the second number" " " stackrel(= 44)overbrace"the resultant number is 44}}}$

Our two equations from the two bits of information are:
$F + S = 12$
$3 F + 5 S = 44$

Now let's change the first equation so that we can solve for one of the variables.
$F + S = 12$
$F = 12 - S$

Now substitute it into the second equation and solve:
$3 F + 5 S = 44$
$3 \left(12 - S\right) + 5 S = 44$
$36 - 3 S + 5 S = 44$
$36 + 2 S = 44$
$2 S = 8$
$S = 4$

Now that we know $S$. substitute it into one of the equations and solve it for F. Either equation would work, but I will use this one:
$F = 12 - S$
$F = 12 - 4$
$F = 8$

CHECK:
$3 F + 5 S = 44$ this should be right if our numbers are correct.

$3 \left(8\right) + 5 \left(4\right) = 44$
$24 + 20 = 44$
$44 = 44$ True, so our numbers are correct.