Question 357762: Solve the following system by the addition method.
1/2x-1/3y= -4
-1/12x+1/10y= 2
x=
y=
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! 1/2x-1/3y= -4
-1/12x+1/10y= 2
Fractions are a pain to work with. So when possible it helps to eliminate them. Fractions can be eliminated from an equation by multiplying both sides by the Lowest Common Denominator (LCD) of all denominators on both sides of the equation. In the first equation the LCD is 6 and in the second equation the LCD is 60. Multiplying these equations by these numbers we get:
6(1/2x-1/3y) = 6(-4)
60(-1/12x+1/10y)= 60(2)
On the left sides we will use the Distributive Property:
6(1/2x)-6(1/3y) = -24
60(-1/12x)+60(1/10y)= 120
which simplifies to:
3x - 2y = -24
-5x + 6y = 120
Now we can more easily continue with the Addition Method. This requires that we have opposite terms in the two equations. There are a variety of ways to create opposites. The simples I see is: "If the first equation is multiplied by 3 then the y terms will be opposites." So that is what I will do:
3(3x - 2y) = 3(-24)
-5x + 6y = 120
Again we use the Distributive Property on the left side:
3(3x) - 3(2y) = -72
-5x + 6y = 120
which simplifies to:
9x - 6y = -72
-5x + 6y = 120
Now that we have opposites lined up, we can add the two equations. (Hence the name of the "Addition Method".) The addition results in the equation:
4x = 48
Dividing by 4 we get:
x = 12
This is only half of the answer. We need the y value, too. To find it we will use the x value and one of the earlier equations. Using x = 12 and the equation
1/2x-1/3y= -4 (It's usually best to use one of the original equations and not one that you have modified.) we get:
(1/2)(12) - (1/3)y = -4
whic simplifies to:
6 - (1/3)y = -4
Subtracting six from each side:
-(1/3)y = -10
Multiplying both sides by -3:
y = 30
So the solution is:
x = 12, y = 30 or (12, 30)
If you want to check the answer, then use the equation that was not used to find y: -1/12x+1/10y= 2
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