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! ______________________________________
Step-by-step explanation:when graphing your solution, you are going to have two rays that will cross if there is one solution, that will be the solution. if they never meet, there is no solution to your problem, but if they are on top of each other, there are infinite solutions.
