In order to solve the system of equations (x-3y=2, 2x+y=11 using the elimination method, which of the following steps could be used? A. Multiply the first equation by 2 and then add that result to the second equationB. Multiply the first equation by 3 and then add that result to the second equationC. Multiply the second equation by 2 and then add that result to the first equationD. Multiply the second equation by 3 and then add that result to the first equation
Accepted Solution
A:
The correct answers are: __________________________________________________________ [C]: "Multiply the second equation by 2 and then add that result to the first equation" ; AND: __________________________________________________________ [D]: "Multiply the second equation by 3 and then add that result to the first equation" . __________________________________________________________ Explanation: __________________________________________________________ Given: __________________________________________________________ The first equation: " x − y = 2 " ;
The second equation: " 2x + y = 11 " ; _____________________________________________________ Consider choice [A]: "Multiply the first equation by 2 and then add that result to the second equation" .
→ Multiply the first equation by "2" :
2 * {x − 3y = 2} ;
→ 2x − 6y = 4 ;
Then, add this to the second equation:
2x − 6y = 4 + 2x + y = 11 _________________ 4x − 5y = 15 ; → RULE OUT "Choice [A]" . ________________________________ Consider choice [B]: "Multiply the first equation by 3 and then add that result to the second equation" ;
→ Multiply the first equation by "3" :
3 * {x − 3y = 2} ;
→ 3x − 9y = 6 ;
Then, add this to the second equation:
3x − 9y = 6 + 2x + y = 11 _________________ 5x − 8y = 17 ; → RULE OUT "Choice [B]" . ____________________________________________________. Consider choice [C]: "Multiply the second equation by 2 and then add that result to the first equation" ;
→ Multiply the SECOND equation by "2" :
2 * (2x + y = 11} ;
→ 4x + 2y = 22 ;
Then, add this to the first equation:
4x + 2y = 22 + x − 3y = 11 _________________ 5x − y = 33 ; → This is a correct answer choice—since we can easily isolate "y" on one side of the equation:
→ 5x − y = 33 ;
↔ -y + 5x = 33 ;
Subtract "5x" from each side of the equation;
→ -y + 5x − 5x = 33 − 5x ;
→ -y = 33 − 5x ;
Multiply each side of the equation by "-1" ; → (-1) * (-y) = (-1) * (33 − 5x) ;
→ y = 5x − 33 ; ____________________________________________________ Consider choice [D]: "Multiply the second equation by 3 and then add that result to the first equation" ;
Divide EACH SIDE of the equation by "7" ; to isolate "x" on one side of the equation; & to solve for "x" ;
7x / 7 = 35/7 ;
x = 5 ; → Yes; this answer choice, [D]; is a correct step.
Furthermore, we can take: "y = 5x − 33 " ; from "choice [D]: and plug in "5" into "x" ; to solve for "y" ;
→ y = 5(5) − 33 = 25 − 33 = -8. ________________________________________________ So, the answer to this system of equations is: ________________________________________________ "x = 5, y = 8 " ; or, write as: " [5, 8] " . ________________________________________________