The numbers are:"9"and "12" . ___________________________________ Explanation: ___________________________________ Let: "x" be the "first number" ; AND:
Let: "y" be the "second number" . ___________________________________ From the question/problem, we are given: ___________________________________ 2x + 5y = 78 ; → "the first equation" ; AND:
5x − y = 33 ; → "the second equation" . ____________________________________ From "the second equation" ; which is:
" 5x − y = 33" ;
→ Add "y" to EACH side of the equation;
5x − y + y = 33 + y ;
to get: 5x = 33 + y ;
Now, subtract: "33" from each side of the equation; to isolate "y" on one side of the equation ; and to solve for "y" (in term of "x");
5x − 33 = 33 + y − 33 ;
to get: " 5x − 33 = y " ; ↔ " y = 5x − 33 " . _____________________________________________ Note: We choose "the second equation"; because "the second equation"; that is; "5x − y = 33" ; already has a "y" value with no "coefficient" ; & it is easier to solve for one of our numbers (variables); that is, "x" or "y"; in terms of the other one; & then substitute that value into "the first equation". ____________________________________________________ Now, let us take "the first equation" ; which is: " 2x + 5y = 78 " ; _______________________________________ We have our obtained value; " y = 5x − 33 " . _______________________________________ We shall take our obtained value for "y" ; which is: "(5x− 33") ; and plug this value into the "y" value in the "first equation"; and solve for "x" ; ________________________________________________ Take the "first equation": ________________________________________________ → " 2x + 5y = 78 " ; and write as: ________________________________________________ → " 2x + 5(5x − 33) = 78 " ; ________________________________________________ Note the "distributive property of multiplication" : ________________________________________________ a(b + c) = ab + ac ; AND:
a(b − c) = ab − ac . ________________________________________________ So; using the "distributive property of multiplication:
→ +5(5x − 33) = (5*5x) − (5*33) = +25x − 165 . ___________________________________________________ So we can rewrite our equation:
→ " 2x + 5(5x − 33) = 78 " ;
by substituting the: "+ 5(5x − 33) " ; with: "+25x − 165" ; as follows: _____________________________________________________
→ " 2x + 25x − 165 = 78 " ; _____________________________________________________ → Now, combine the "like terms" on the "left-hand side" of the equation:
+2x + 25x = +27x ;
Note: There are no "like terms" on the "right-hand side" of the equation. _____________________________________________________ → Rewrite the equation as: _____________________________________________________ → " 27x − 165 = 78 " ;
Now, add "165" to EACH SIDE of the equation; as follows:
→ 27x − 165 + 165 = 78 + 165 ;
→ to get: 27x = 243 ; _____________________________________________________ Now, divide EACH SIDE of the equation by "27" ; to isolate "x" on one side of the equation ; and to solve for "x" ; _____________________________________________________ 27x / 27 = 243 / 27 ;
→ to get: x = 9 ; which is "the first number" . _____________________________________________________ Now; Let's go back to our "first equation" and "second equation" to solve for "y" (our "second number"):
2x + 5y = 78 ; (first equation);
5x − y = 33 ; (second equation); ______________________________ Start with our "second equation"; to solve for "y"; plug in "9" for "x" ;
→ 5(9) − y = 33 ;
45 − y = 33;
Add "y" to each side of the equation:
45 − y + y = 33 + y ; to get:
45 = 33 + y ;
↔ y + 33 = 45 ; Subtract "33" from each side of the equation; to isolate "y" on one side of the equation ; & to solve for "y" ;
→ y + 33 − 33 = 45 − 33 ;
to get: y = 12 ;
So; x = 9 ; and y = 12 . The numbers are: "9" and "12" . ____________________________________________ To check our work: _______________________ 1) Let us plug these values into the original "second equation" ; to see if the equation holds true (with "x = 9" ; and "y = 12") ;
→ 5x − y = 33 ; → 5(9) − 12 =? 33 ?? ; → 45 − 12 =? 33 ?? ; Yes! ________________________ 2) Let us plug these values into the original "second equation" ; to see if the equation holds true (with "x = 9" ; and "y = 12") ;
→ 2x + 5y = 78 ; → 2(9) + 5(12) =? 78?? ; → 18 + 60 =? 78?? ; Yes! _____________________________________ So, these answers do make sense! ______________________________________
