Solve by Elimination

Question

Solve by elimination:

 

10x-8y=4

-5x+3y=-9

 

also

 

-3x-10y=4

x-5y=18

Answer 1

a)

  10x-8y=4    [eq a1]
  -5x+3y=-9    [eq a2]

To solve by elimination, we need one set of coefficients (either x-coefficients or y-coefficients) to be opposite or the same, so we can eliminate them (by addition or subtraction).

For this problem, let's pick the x term:

     20x - 16y =    8        [multiply eq a1 by 2]

    -20x +12y = -36        [multiply eq a2 by 4]

  ----------------------      [elimination; add equations]

              -4y  = -28

                 y  =    7       [eq a3]

 

               8y   =  56      [multiply eq a3 by 8]

       10x - 8y   =   4       [eq a1 again]

   ---------------------      [elimination; add equations]

       10x          =  60

          x          =   6

 

Checking (very important):

  Is  10*6 -8*7 = 4   ?

          60 - 56 = 4   ?

                4    = 4   ?yes

  Is  (-5)*6 + 3*7 = -9   ?

           -30  +  21  = -9   ?

                  -9  =  -9   ?yes

 

b)

  -3x-10y= 4   [eq b1]
     x- 5y=18   [eq b2]

Let's eliminate y this time --

   -3x  -10y =    4    [eq b1 again]

   -2x +10y = -36    [multiply eq b2 by (-2)]

 --------------------    [elimination; add equations]

   -5x          = -32

      x          = 32/5

 

Special note:  You may proceed and solve for y, but I think that there was a typo and that equation b1 should have been:  -3x -10y =-4.  I will solve that system:

   -3x -10y =    -4

   -2x +10y = -36

---------------------

   -5x         = -40

      x         = 8

 

      x    -5y=18      [elimination; subtract this eq from the line above]

  -----------------

             5y = -10

               y = -2

 

Now, you should check that answer (8,-2) and double check whether there was a typo.  If no typo, then you know how to solve by elimination, right?

Answer 2

Alright, for the first set of equations, we have these two equations:

 

10x-8y=4
-5x+3y=-9

 

To use elimination to solve, we can multiply the bottom equation by two to get:

 

2(-5x + 3y = -9)

-10x + 6y = -18

 

Now we can add the first equation and this equation together to get:

 

    10x  - 8y =    4

+ -10x + 6y = -18

--------------------------

           -2y  =  -14

 

Now just divide both sides by -2 to get:

y = 7

Now, to solve for x, just plug this into any of the other equations.  We'll use the first equation:

10x-8y=4

10x-8(7)=4

10x-56=4

10x = 60

x = 6

So for the first set of equations, x = 6 and y = 7

 

For the second set of equations, we have these equations:

-3x-10y=4
x-5y=18

 

Let's multiply the bottom equation by 3 to get:

3(x-5y=18)

3x - 15y = 54

 

Now lets add this equation and the first equation:

     -3x - 10y =   4

+    3x - 15y = 54

------------------------

            -25y = 58

Divide both sides by -25 to get:

y = -(58/25) = -2.32

Now lets plug y into one of the equations to find x.  Let's use the second equation:

x-5y=18

x - 5(-2.32) = 18

x - (-11.6) = 18

x + 11.6 = 18

x = 6.4

So our solutions to the second set of equations is x = 6.4 and y = -2.32

 

 

 

Answer 3

-5x + 3y = -9

=> 5x = 3y + 9

 

Plugging above in the other equation

 

=> 2(5x) - 8y = 4

=> 2(3y + 9) - 8y = 4

=> -2y + 18 = 4

=> y = 7

 Plugging value of y

=> x = 0.2( 3(7) + 9 ) = 6

 

 

x - 5y = 18
=> x = 5y + 18

Plugging above in the other equation

=> -3(x) - 10y = 4
=> -3(5y + 18) - 10y = 4
=> -25y - 54 = 4
=> y = -58/25
Plugging value of y
=> x = 5( -58/25 ) + 18 = 32/5