Solve by the addition method

Question

LaTeX: 5x+15y=20
5
x
+
15
y
=
20

LaTeX: 2x+6y=8

Answer 1

the system:
x + 3 y = 4, (eq. 3)
2x +6y = 8. (eq. 2)

 

The lower equation is a multiple of the top equation.

Their lines are collinear.

All solution to one are solutions to the other.

 

Answer 2

5x+15y=20      (eq. 1)

2x+6y=8          (eq. 2)

 

We divide (eq. 1) by 5:

(5x + 15 y)/5 = 20/5

x    + 3 y        = 4    (eq. 3)

 

the system:

x + 3 y = 4,    (eq. 3)

2x +6y = 8.    (eq. 2)

 

We multiply (eq. 3) by 2 and then subtract (eq. 2)

(x + 3 y) 2 = 4 x 2,

2x + 6 y     = 8           (eq. 4)

 

  2x + 6 y = 8               (eq. 4)

-

  2x + 6 y = 8               (eq. 2)

 0 x  + 0 y = 0

           0 = 0                TRUE statement (Zero is equal to zero).

 

This means, that the system above has infinitely many solutions.

All points on the line 2x + 6y = 8.

 

As an ordered pair of numbers (x, (8-x)/3) describes all the solutions.