SOLUTION: 1.Solve x+y=a+b , ax-by=a2-b2 2.Solve graphically system of equation 2x-y=5; 4x-y=8

1.Solve

  x +  y = a  + b 
 ax - by = a² - b² 

Eliminate y

Multiply the first equation thru by b

 bx + by =  ab + b² 
 ax - by =  a² - b² 

Now add them vertically:

  bx + by =  ab  + b² 
  ax - by =  a²  - b² 
——-—————————-———————————
bx+ax    =  ab+a²

   bx + ax = ab + a²

Factor out x on the left, factor out a on the right

  x(b + a) = a(b + a)



Divide both sides by (a + b)

  x(b + a)     a(a + b)
 —————————— = ——————————
   (a + b)      (a + b)

 
Cancel

  x(b + a)     a(a + b)
 —————————— = ——————————
   (a + b)      (a + b)

             
          x = a   
  
Now eliminate x

Multiply the first equation thru by -a

-ax - ay = -a² - ab 
 ax - by =  a² - b² 

Now add them vertically:

-ax -  ay   =  -a² - ab 
 ax -  by   =   a² - b² 
——-—————————-———————————
    -ay-by =       -ab-b²

Multiply thru by -1

   ay + by = ab + b²

Factor out y on the left, factor out b on the right

  y(a + b) = b(a + b)

Divide both sides by (a + b)

  y(a + b)     b(a + b)
 —————————— = ——————————
   (a + b)     (a + b)

 
Cancel

  y(a + b)     b(a + b)
 —————————— = ——————————
   (a + b)     (a + b)   
 

          y = b 

2.Solve graphically system of equation 2x-y=5; 4x-y=8

Plot some points for 2x - y = 5

 x |  y 
 ------
 1 | -3
 2 | -1
 3 |  1
 4 |  3

Plot some points for 4x - y = 8

 x |  y 
 ------
 1 | -4
 2 |  0
 3 |  4

 

Notice that the point where the two lines intersect looks like it is
exactly to the right of -2 on the y-axis.  It also look like it is
directly below 1 1/2 on the x-axis. So I would guess that the coordinates
of that point where they cross is (1.5, -2) so the answers read from the
way the graph looks are x = 1.5, y = -2

Edwin
AnlytcPhil@aol.com