The numbers are: "3" and "9" . _______________________________________________________ Explanation: _______________________________________________________ Let "x" represent one of the two (2) numbers.
Let "y" represent the other one of the two (2) numbers.
x = 2y + 3 ;
x + y = 12 . __________________________ Method 1) __________________________ x = 12 <span>− y ;
Plug this into "x" for "2y + 3 = x" ;
</span>→ 2y + 3 = 12 <span>− y ; </span> Add "y" to each side of the equation; & subtract "3" from each side of the equation ;
→ 2y + 3 + y − 3 = 12 − y + y <span>− 3 ; </span> to get: 3y = 9 ;
Divide each side of the equation by "3" ; to isolate "y" on one side of the equation; & to solve for "y" ;
3y / 3 = 9 / 3 ;
y = 3 . ____________________________ Now: x = 12 − y ; Plug in "3" for "y" ; to solve for "x" ;
→ x = 12 − 3 = 9 ____________________________ So; x = 9, y = 3 . ____________________________ Method 2) ____________________________ When we have: ____________________________ x = 2y + 3 ;
x + y = 12 . ____________________________ → y = 12 − x ; _____________________________ Substitute "(12−x)" for "y" in the equation:
" x = 2y + 3 " ;
→ x = 2(12 − x) + 3 ; _____________________________________ Note the "distributive property of multiplication" : _____________________________________ a(b + c) = ab + ac ;
a(b − c) = ab − ac ; _____________________________________ As such: _____________________________________ → 2(12 − x) = 2(12) − 2(x) = 24 − 2x ; _____________________________________ So; rewrite: _____________________________________ x = 2(12 − x) + 3 ; as: _____________________________________ → x = 24 − 2x + 3 ;
→ x = 27 − 2x ;
Add "2x" to each side of the equation:
→ x + 2x = 27 − 2x + 2x ;
→ 3x = 27 ;
Divide each side of the equation by "3" ; to isolate "x" on one side of the equation; & to solve for "x" ;
3x / 3 = 27 / 3 ;
x = 9 . ___________________________ Note: "y = 12 − x" ; Substitute "9" for "x" ; to solve for "y" ;
→ y = 12 − 9 = 3 ;
→ y = 3 . __________________________ So, x = 9 ; and y = 3. ____________________________ The numbers are: "3" and "9" . _____________________________ To check our answers: Let us plug these numbers into the original equations; to see if the equations hold true ; (i.e. when, "x = 9" ; and "y = 3" _____________________________ → x + y = 12 ;