Three times the larger of two numbers is equal to four times the smaller. the sum of the numbers is 21. find the numbers.

X= smaller number
y= larger number

We will have two equations to work with so we can solve for the variables.

EQUATION 1:
3 * y= 4 * x
3y= 4x
subtract both sides by 3y
0= 4x - 3y

EQUATION 2:
21= x + y

SOLVE:
solve for one variable in equation 2
21= x + y
subtract both sides by y
21-y= x

Substitute x= 21-y in equation 1:

0= 4x - 3y
0= 4(21-y) - 3y
0= (4*21) + (4*-y) - 3y
0= 84 - 4y - 3y
0= 84 - 7y
add 7y to both sides
7y= 84
divide both sides by 7
y= 12

Substitute y=12 in either equation to solve for x:

21= x + y
21= x + 12
subtract both sides by 12
9= x

CHECK:
0= 4x - 3y
0= 4(9) - 3(12)
0= 36 - 36
0= 0

ANSWER: x=9 and y=12

Hope this helps! :)