How to find x and y algebraically?

Question

-7x+5y=0

14x-8y=2

Answer 1

You may solve this system of equations algebraically using either (1) substitution, (2) elimination, (3)[other methods].  Other tutors have used substitution; I'll use elimination.

 

To eliminate one variable by either adding or subtracting equations, make the corresponding coefficients on that variable either the same or opposite.  For this problem, the coefficients of x may be made opposite by multiplying the first equation by 2:

   -14x + 10y = 0           [2 * first equation]

    14x  -   8y  = 2          [second equation]

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

                2y  = 2

                  y  = 1         [divide both sides by 2; this is equation 3]

 

                 8y  = 8         [8 * equation 3 to match coefficient of y]

     14x  -   8y   = 2         [second equation again]

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

     14x             = 10

         x            = 10/14     [divide both sides by 14]

          x           =  5/7        [reduce fraction]

 

Checking (very important):

  Is    -7(5/7) + 5(1) = 0    ?

            -5      +  5    =  0

                    0          =  0   ? yes

  Is    14(5/7) - 8(1) = 2   ?

             10    -  8    = 2   ?

                    2        =  2   ? yes

Answer 2

-7x+5y=0

14x-8y=2

the simplest thing to do is multiply the first equation by 2

2(-7x)+2(5y)=2(0)

-14x+10y=0

14x-8y=2

add

2y=2

y=1

substitute into either original equation

-7x+5(1)=0

-7x+5=0

5=7x

x=5/7

(5/7, 1) is the solution

Answer 3

First, solve one of the equations for one of the variables.  It doesn't matter which.  For example:

-7x+5y=0

5y=7x

y=(7x)/5

y=1.4x

Now substitute the value for y into the other equation:

14x-8y=2

14x-(8*1.4x)=2

14x-11.2x=2

2.8x=2

x=2/2.8

x=.7143

Now substitute the value for x into one of the equations and solve for y:

14*0.7143 -8y = 2

10 - 8y = 2

-8y = -8

y = 1

 

Answer 4

y=1 and x=5/7

 

you basically isolate one of the variables in one of the equations. You substitute that answer into the other equation and solve for that variable. 

 

14x-8y=2

-8y=2-14x

y=(14x-2)/8

-7x+5y=0

-7x+5(14x-2)/8=0

-7x+(70x-10)/8=0

=-7x+10(7x-1)/8=0

=-7x+5(7x-1)/4=0

=-28x+35x-5=0

=7x-5=0

=7x=5

x=5/7

 

now substitute that answer into the other equation and you will find y=1

-7(5/7)+5(y)=0

-5+5y=0

5y=5

y=1