# The sum of two numbers is 42. The smaller number is 6 less than the larger number. What are the numbers?

The sum of two numbers is 42. The smaller number is 6 less than the larger number. What are the numbers? 24 and 18

Answer: The 2 (TWO) numbers are: 18 and 24 . ______________________________________________ Explanation: ______________________________________________ x + y = 42 ; Let "x" be the 'smaller number' ; "y'' be the "larger number"; so: x = y − 6 ; Now, substitute "y−6" for "x" in the equation: _____________________________________ x + y = 42 ; _________________________________________________ y − 6 + y = 42 _________________________________________________ Combine the "like terms", which appear on the left-hand side of the equation: __________________________________________________ y + y , = 2y ; and rewrite the equation: __________________________________________________ 2y − 6 = 42 ; __________________________________________________ Now, add "6" to each side of the equation: __________________________________________________ 2y − 6 + 6 = 42 + 6 ; __________________________________________________ to get: 2y = 48 ; __________________________________________________ Now, divide EACH side of the equation by "2"; to isolation "y" on one side of the equation; and to solve for "y" ; ___________________________________________________ 2y / 2 = 48 / 2 ; ___________________________________________________ to get: y = 24 ; which is the value of one of the number we which to solve for —specifically, the larger number. ____________________________ Now, we can use one of two methods to solve for "x". In fact, let us use BOTH methods, to ensure we have the same value for "x" (which serves a purpose for confirming our answer and checking our work). We can start with EITHER method, first in EITHER ORDER; nonetheless, I shall still list them as "Method 1" and "Method 2"—as follows: _______________________________ Method 1) From the original problem, "The smaller number is 6 less than the smaller number" ; or: " x = y − 6 " . ______________________________________ Since our obtained/ solved value for "y" is "24" ; we can plug in the value, "24", for "y"; into this equation; and solve for "x" ; _____________________________________________ x = y − 6 = 24 - 6 ; ___________________________ to get: x = 18 . which is the other value we wish to obtain, the "smaller number" value. ______________________________ Method 2) From the original problem: "The sum of two numbers is 42. ...." ; ______________________________________________ or, " x + y = 42 " . ______________________________________________ Using our obtained/solved value for "y", which is: "24", we can plug this value, "24", for "y", into the above equation; and solve for "x"; _________________________________________________ x + y = 42 ; _________________________________________________ x + 24 = 42 ; _________________________________________________ Now, subtract "24" from EACH side of the equation, to isolate "x" on one side of the equation; and to solve for "x" ; _________________________________________________ x + 24 − 24 = 42 − 24 ; _________________________________________________ x = 18 ; which is the same value we got for "x" from: "Method 1" above. _____________________________ So, the two numbers are: "18" and "24". Do they add up to "42"? 18 + 24 =? 42?? Yes. Is the "smaller number" (18) , "6 less than the larger number (24)? Does 18 + 6 =? 24 ?? Yes! Does 24 − 6 =? 18 ?? Yes! ___________________________________________________ So the two (2) numbers are: 18 and 24 . ____________________________________________