The answer is: [A]: -2x + 3y = 21 .
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Note that "standard form" refers to:
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→ 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).
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I shall demonstrate 2 (two) methods to convert the equation given:
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→ y = (⅔) x + 7 ; to "standard form".).
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Method 1:
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→ Given: y = (⅔)x + 7 ;
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→ 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 " ;
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→ y = (⅔)x + 7 ; → y − y − 7 = (⅔)x + 7 − y − 7 ; To get:
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→ -7 = (⅔)x − y ↔ Rewrite as: (⅔)x − y = -7 ;
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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 "1y = y;
since: y1 = 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.
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However, this equation—as written—does NOT match any of the answer choices given.
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→So, now we should multiply our ENTIRE equation (BOTH SIDES) by "-3"; for the sake of:
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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" ;
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→ {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}.
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We have: (⅔)x − y = -7 ; We shall multiply EACH SIDE of the equation by "-3"; → -3 {(⅔)x − y = -7} ;
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→ Note: Start with:
→"(-3 (⅔)x = -3 ⅔ x ;
→ -3 ⅔ = \(\frac{-3}{1}\) \(\frac{2}{3}\) =(-32)/(13)
= -6 / 3 = -2 ; → Don't forget to bring down the "x": -2 x = -2x ;
→ So: - 3 (⅔)x = - 2x ;
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The next term: -3 -1y = +3y ; (Note: -3 -1y = (-3 -1)y = 3y = +3y ;
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The next term: -3(-7) = +21 ;
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→ to get: → -2x + 3y = 21 ; which is: 'Answer choice: [A]'.
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Method 2:
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Given: y = (⅔)x + 7 ; write in standard form:
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→ Given: y = (⅔)x + 7 ; Multiply EACH SIDE of the equation by "3", to get rid of the fraction, "(⅔)" ;
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→ 3 { y = (⅔)x + 7 } ; to get:
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First term: 3y = 3y;
Second term: 3((⅔)x)) = 3 (⅔) x ; 3 (⅔) = (3/1)(2/3) =(32)/(13) =
6/3 = 2; → Don't forget the "x" → 2 x
= 2x;
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Third term: 37 = 21;
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→To get: → 3y = 2x + 21 ;
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→ Now, subtract "3y" and subtract "21" from EACH SIDE of the equation:
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→ 3y = 2x + 21 ; → 3y − 21 − 3y = 2x + 21 − 3y − 21 ;
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to get: → -21 = 2x − 3y ; → Rewrite as: → 2x − 3y = -21 ;
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Note: This would appear in "standard form" equation; "Ax + By = C";
in which A = 2; B = 3; C = -21;
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However, " 2x − 3y = -21 " ; is NOT an answer choice given.
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However, there are answer choices given with "21" {note: "positive 21"} representing the number for "C" within the "standard form" equation:
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→ "Ax + By = C"
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So, given our equation:
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→ 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:
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→ -1 {2x − 3y = -21} ;
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First term: -1 2x = -2x ;
Second term: -1 -3y = -1 -3 y = +3y ;
Third term: -1 * -21 = + 21
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→ -2x + 3y = 21 ; which is: 'Answer choice: [A]'.
