SOLUTION: Solve the system by addition. 2x – 4y = 7 4x + 2y = 9

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Question 69844: Solve the system by addition.
2x – 4y = 7
4x + 2y = 9
Found 2 solutions by bucky, Edwin McCravy:
Answer by bucky(2189)   (Show Source): You can put this solution on YOUR website!
2x – 4y = 7
4x + 2y = 9
Note that you can't solve this pair of equations by just adding them together because
there isn't a term in the first equation that is equal to a corresponding term in the
second equation but that has the opposite sign so that when they are added they cancel.
You must first do something to one of the equations that make a term in each equation
have the same term but also have opposite signs. For example, in this problem, what would
happen if you multiplied all the terms of the bottom equation by 2. The pair of equations
would then be:
2x – 4y = 7
8x + 4y = 18
Now you can see that the term 4y appears in both equations, but it has a minus sign in
the first equation and a plus sign in the second equation. Now if you add the two
equations vertically (this satisfies the need to solve it by addition) the 4y terms
cancel and the equation you end up with is:
10x = 25
Dividing both sides by 10 results in:
x = 2.5
You can now solve for y by substituting 2.5 for x in either of the two original equations.
Let's use the first equation.
2x-4y = 7
Put 2.5 in place of x to get:
2(2.5) - 4y = 7
Multiply out 2(2.5) and the equation becomes:
5 - 4y = 7
Subtract 5 from both sides and the equation reduces to:
-4y = 2
Finally divide both sides by -4 and you are left with:
y = -0.5
That solves the problem ... x = 2.5 and y=-0.5 are the answers.
Is that the only way to solve this problem by addition? Nope. Way back at the beginning
we could have multiplied the top equation by -2. This would have made the pair of
equations:
-4x +8y = -14
4x + 2y = 9
You can now solve this pair by addition because the minus 4x will cancel with the plus 4x.
Try it. You should get the same answers for x and y as we did above.
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!

To find y we eliminate x:

To eliminate the x terms, look at the coefficients
of x:

2x – 4y = 7
4x + 2y = 9

They are 2 and 4.   Their least common multiple in
absolute value is 4. Multiply the first equation 
through by -2 to make the 2x become -4x, so it will 
cancel with the 4x in the second equation:

-2[2x - 4y = 7]
   4x + 2y = 9 

  -4x + 8y = -14
   4x + 2y =   9

Now add vertically term-by-term:

  -4x + 8y = -14
   4x + 2y =   9
       10y =  -5
         y = -5/10
         y = -1/2


To find x we eliminate y:

To eliminate the y terms, look at the coefficients
of y:

2x – 4y = 7
4x + 2y = 9

They are -4 and 2.   Their least common multiple in
absolute value is 4. Multiply the second equation 
through by 2 to make the 2y become 4y, so it will 
cancel with the -4y in the first equation:

   2x - 4y = 7
 2[4x + 2y = 9] 

   2x - 4y =  7
   8x + 4y = 18

Now add vertically term-by-term:

   2x - 4y =  7
   8x + 4y = 18
  10x      = 25
         x = 25/10
         x = 5/2

So the solution is (x,y) = (5/2,-1/2) 

Edwin
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