The answer is: [A]: -2x + 3y = 21 .
___________________
Note that "standard form" refers to:
_______________________________
→ Ax +By =C ;
in which "x" and "y" are variables; "A" and "B" represent the coefficients before those variables, respectively; and "C" is a number (not a variable).
_______________________________
I shall demonstrate 2 (two) methods to convert the equation given:
________________________________________
→ y = (⅔) x + 7 ; to "standard form".).
_____________________
Method 1:
_______________________
→ Given: y = (⅔)x + 7 ;
______________________
→ Subtract "y" from EACH SIDE of the equation; and subtract "7" from EACH SIDE of the equation. Doing so: 1) puts the "x" and "y' values on one side of the equation; 2) removes a numeric value from the side of the equation with the "x" and "y" values; and 3) puts a numeric value on a side of the equation that does not have any "x" or "y" values — all three of which help to convert to equation to "standard form"—that is, " Ax + By = C " ;
____________
→ y = (⅔)x + 7 ; → y − y − 7 = (⅔)x + 7 − y − 7 ; To get:
____________
→ -7 = (⅔)x − y ↔ Rewrite as: (⅔)x − y = -7 ;
________________________
Note: While this very would could be an answer in"standard form";
"Ax + By = C", in which A = ⅔ , B = -1; and C = -7;
(Note: There is an implied "1" before the "y", since "1*y = y;
since: y*1 = y; (anything * 1 = said number);
and the coefficient is "-1" (NEGATIVE 1); since the "standard form" is: "Ax + By = C; and we have:
(⅔)x − y = -7 ; so the:
"− y" functions as a: PLUS "negative 1y"; in this circumstance.
_________________________________________________
However, this equation—as written—does NOT match any of the answer choices given.
___________________
→So, now we should multiply our ENTIRE equation (BOTH SIDES) by "-3"; for the sake of:
___________________
1) getting an answer choice that matches one of the answer choices given;
AND:
2) for the general sake of simplicity—which would include getting the "-7" changed to a "positive 21" ;
_______________________.
→ {Note: A non-zero, negative integer, when multiplied by another non-zero, negative integer; will result in a non-zero POSITIVE integer.
As such: -7 *-3 = 21}.
_________________
We have: (⅔)x − y = -7 ; We shall multiply EACH SIDE of the equation by "-3"; → -3 * {(⅔)x − y = -7} ;
______________________
→ Note: Start with:
→"(-3 * (⅔)x = -3 * ⅔ * x ;
→ -3 * ⅔ = [tex] \frac{-3}{1} [/tex] * [tex] \frac{2}{3} [/tex] =(-3*2)/(1*3)
= -6 / 3 = -2 ; → Don't forget to bring down the "x": -2 * x = -2x ;
→ So: - 3 *(⅔)x = - 2x ;
_________________
The next term: -3 *-1y = +3y ; (Note: -3 *-1y = (-3* -1)*y = 3*y = +3y ;
____________________
The next term: -3*(-7) = +21 ;
_______________________________________
→ to get: → -2x + 3y = 21 ; which is: 'Answer choice: [A]'.
___________________________________________________________
Method 2:
____________
Given: y = (⅔)x + 7 ; write in standard form:
_________________________
→ Given: y = (⅔)x + 7 ; Multiply EACH SIDE of the equation by "3", to get rid of the fraction, "(⅔)" ;
___________________________
→ 3* { y = (⅔)x + 7 } ; to get:
____________________________
First term: 3*y = 3y;
Second term: 3*((⅔)x)) = 3 * (⅔) * x ; 3 * (⅔) = (3/1)*(2/3) =(3*2)/(1*3) =
6/3 = 2; → Don't forget the "x" → 2 * x
= 2x;
___________________
Third term: 3*7 = 21;
__________________
→To get: → 3y = 2x + 21 ;
____________________________
→ Now, subtract "3y" and subtract "21" from EACH SIDE of the equation:
____________________________
→ 3y = 2x + 21 ; → 3y − 21 − 3y = 2x + 21 − 3y − 21 ;
_____________________________________________
to get: → -21 = 2x − 3y ; → Rewrite as: → 2x − 3y = -21 ;
______________________________
Note: This would appear in "standard form" equation; "Ax + By = C";
in which A = 2; B = 3; C = -21;
___________________________________________
However, " 2x − 3y = -21 " ; is NOT an answer choice given.
______________________________
However, there are answer choices given with "21" {note: "positive 21"} representing the number for "C" within the "standard form" equation:
________
→ "Ax + By = C"
_________________
So, given our equation:
________________________
→ 2x − 3y = -21 ;
→ We can multiply the ENTIRE equation (BOTH sides) by "-1" ; to change the "-21" value to "21" (positive 21); and see if the equation matches any of the answer choices:
______________
→ -1 * {2x − 3y = -21} ;
______________
First term: -1* 2x = -2x ;
Second term: -1 * -3y = -1 * -3 * y = +3y ;
Third term: -1 * -21 = + 21
___________________________
→ -2x + 3y = 21 ; which is: 'Answer choice: [A]'.
