Question 203365: Could you please help me out with this problem??
Solve by elimination method
7/2x + 7/3y = 42
1/4x + 1/3y = 4
Thank you very much!!
Found 2 solutions by stanbon, jsmallt9:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Solve by elimination method
7/2x + 7/3y = 42
1/4x + 1/3y = 4
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Multiply thru the 1st by 6; Multiply thru the 2nd by 12
21x + 14y = 252
3x + 4y = 48
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Solve the 2nd for "x"
x = (-4/3)y + 16
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Substitute into the modified 1st equation:
21((-4/3)y + 16) + 14y = 252
-28y + 336 + 14y = 252
-14y = - 84
y = 6
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Substitute into x = (-4/3)y + 16 to get x = (-24/3)+16 = 8
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Ans: (8,6)
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Cheers,
Stan H.
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website! Solve by elimination method
7/2x + 7/3y = 42
1/4x + 1/3y = 4
We can solve with or without the fractions. With the fractions:
Multiply the second equation by -7. Why the second equation and why -7? See what happens and you'll see why.
7/2x + 7/3y = 42
-7/4x + -7/3y = -28
Can you see why we did what we did? We now have opposites lined up. The y terms are opposites. Now if we add the equations together the y terms will cancel each other out and "disappear". This is the key to the elimination method. Somehow you need to get opposites lined up, for the x terms or the y terms. (It doesn't matter which. Look for the easiest way to get the opposites.) Sometimes you need to multiply both equations, one by one number and the other equation by some other number, in order to get the opposites lined up.
There are many ways to get opposites. For example, if we had multiplied the second equation by -14 instead of -7 we would have ended up with opposite x terms!
Finding these opposites are more difficult with the fractions so you may want to see below where we eliminate the fractions before we get the opposites.
Add the equations once you have the opposites lined up. Since we have coefficients which are fractions the addition of the x terms (which are not opposites) is more difficult. (Yet another reason to learn how to eliminate the fractions at the start.) Adding we get:
7/2x + 7/3y = 42
+ -7/4x + -7/3y = -28
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(7/2 + -7/4)x = 14
Since 7/2 + -7/4 = 7/4 we now have:
Solve the equation you get after adding for the remaining variable. Divide both sides by 7/4 (or multiply both sides by its reciprocal, 4/7) I prefer the reciprocal:
Use this solution to find the value for the other variable using either of the original equations. Using the first equation:
Our solution: (8, 6)
Without the fractions.
7/2x + 7/3y = 42
1/4x + 1/3y = 4
One can eliminate the fractions at the start. The "price" of avoiding the fractions is an extra step and larger numbers. Start by multiplying each equation the the Lowest Common Denominator (LCD) of all the denominators it has, including a denominator on the right side if there is one!
The LCD of the first equation is 6 so we multiply both sides of it by 6. The LCD of the second equation is 12 so we will mutliply it by 12:
6(7/2x + 7/3y) = 6(42)
12(1/4x + 1/3y) = 12(4)
This results in:
21x + 14y = 252
3x + 4y = 48
The fractions are gone. In return we have larger numbers. Now we continue with the Elimination Method. First, create opposites. One way to do so would be to multiply both sides of the second equation by -7:
21x + 14y = 252
-21x + -28y = -336
Add the equations:
21x + 14y = 252
+ -21x + -28y = -336
------------------
-14y = -84
Solve this equation. Divide by -14 giving .
Use this to find the value of the other variable. Even though the original equations have fractions, it is recommended to use an original equation to find the other variable. (If you made a mistake when you got rid of the fractions and then you used an incorrect equation, you would not be able to get the correct answer. So to find x:
To check you solution to a system of equations, no matter which method is used, you substitute both variables into both equations and see if they work out. (I have checked and (6, 8) is correct.)
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