How do solve the following linear system?: -6x+6y=6 , -6x+3y=-12 ?

2 Answers
Apr 4, 2018

(x,y) = (5,6)

Explanation:

6y=6x+6
y=x+1 solve for y in terms of x in the first equation
-6x+3(x+1)=-12 plug into the other equation
-6x+3x+3=-12 distribute
3-3x=-12 combine like terms
-3x=-15 subtract 3 from both sides
x=5 divide both sides by -3

Now, going back to the other equation
-6(5)+6y=6 plug in the value 5 for x
-30+6y=6 multiply
6y=36 subtract 30 from both sides
y=6 divide both sides by 6

Apr 4, 2018

Subtract equation 2 from equation 1 to solve for y, and then back-solve for x to find that x=5 and y=6

Explanation:

Let's use linear algebra to solve the system of equations. This will involve subtracting whole equations from each other:

-6x+6y=6
ul(-(-6x+3y=-12))

-6x+6y=6
ul(+6x-3y=12)
cancel(0x)+3y=18

Now that x is eliminated, we can solve for y:

3y=18

rArr (cancel(3)y)/cancel(color(red)(3))=18/color(red)(3)

rArr color(blue)(y=6)

Finally, we can plug our solution for y back into either equation to solve for x:

-6x+3y=-12

rArr -6x+3(color(blue)(6))=-12

rArr -6x+18=-12

rArr -6xcancel(+18color(red)(-18))=-12color(red)(-18)

rArr -6x=-30

rArr (cancel(-6)x)/cancel(color(red)(-6))=(-30)/color(red)(-6)

rArr color(green)(x=5)