SOLUTION: Solve each system by the addition method. 3/7x+5/9y=27 1/9x+2/7y=7 Thanx

Algebra ->  Linear-equations -> SOLUTION: Solve each system by the addition method. 3/7x+5/9y=27 1/9x+2/7y=7 Thanx      Log On


   

Question 177974: Solve each system by the addition method.
3/7x+5/9y=27
1/9x+2/7y=7
Thanx
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition
TEST

%283%2F7%29%2Ax%2B%285%2F9%29%2Ay=27 Start with the first equation


63%28%283%2F7%29%2Ax%2B%285%2F9%29%2Ay%29=%2863%29%2A%2827%29 Multiply both sides by the LCD 63



27%2Ax%2B35%2Ay=1701Distribute and simplify


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%281%2F9%29%2Ax%2B%282%2F7%29%2Ay=7 Start with the second equation


63%28%281%2F9%29%2Ax%2B%282%2F7%29%2Ay%29=%2863%29%2A%287%29 Multiply both sides by the LCD 63



7%2Ax%2B18%2Ay=441 Distribute and simplify



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Lets start with the given system of linear equations

27%2Ax%2B35%2Ay=1701
7%2Ax%2B18%2Ay=441

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 27 and 7 to some equal number, we could try to get them to the LCM.

Since the LCM of 27 and 7 is 189, we need to multiply both sides of the top equation by 7 and multiply both sides of the bottom equation by -27 like this:

7%2A%2827%2Ax%2B35%2Ay%29=%281701%29%2A7 Multiply the top equation (both sides) by 7
-27%2A%287%2Ax%2B18%2Ay%29=%28441%29%2A-27 Multiply the bottom equation (both sides) by -27


So after multiplying we get this:
189%2Ax%2B245%2Ay=11907
-189%2Ax-486%2Ay=-11907

Notice how 189 and -189 add to zero (ie 189%2B-189=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%28189%2Ax-189%2Ax%29%2B%28245%2Ay-486%2Ay%29=11907-11907

%28189-189%29%2Ax%2B%28245-486%29y=11907-11907

cross%28189%2B-189%29%2Ax%2B%28245-486%29%2Ay=11907-11907 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

-241%2Ay=0

y=0%2F-241 Divide both sides by -241 to solve for y



y=0 Reduce


Now plug this answer into the top equation 27%2Ax%2B35%2Ay=1701 to solve for x

27%2Ax%2B35%280%29=1701 Plug in y=0


27%2Ax%2B0=1701 Multiply



27%2Ax=1701-0 Subtract 0 from both sides

27%2Ax=1701 Combine the terms on the right side

cross%28%281%2F27%29%2827%29%29%2Ax=%281701%29%281%2F27%29 Multiply both sides by 1%2F27. This will cancel out 27 on the left side.


x=63 Multiply the terms on the right side


So our answer is

x=63, y=0

which also looks like

(63, 0)

Notice if we graph the equations (if you need help with graphing, check out this solver)

27%2Ax%2B35%2Ay=1701
7%2Ax%2B18%2Ay=441

we get



graph of 27%2Ax%2B35%2Ay=1701 (red) 7%2Ax%2B18%2Ay=441 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (63,0). This verifies our answer.