Identify each system as having one solution, no solution, infinite solutions. 5x-y=9, y=5x+19 and y=6x+14, 12x-2y=-28

Question

How would I do this & identify what solution it has?? Please use step by step, Thank you. 

Answer 1

5x-y=9, y=5x+19  write both equations in same form

y=5x-9 for the first and have

y=5x+19  for the second. These are two parallel lines with different intercepts.  They do not intersect or coincide.  No solutions

 

and y=6x+14, 12x-2y=-28  write both equations in same form

y=6x+14

2y=12x+28  or y=6x+14.  They are the same line.  The solution is all of the points on the line.

Answer 2

Hi again Deanna;

5x-y=9

y=5x+19

This definitely has NO solution...

Let's begin with the first equation...

5x-y=9

Let's isolate y by subtracting 5x from both sides...

-x+5x-y=-5x+9

-y=-5x+9

Let's multiply both sides by -1...

(-1)(-y)=-1(-5x+9)

y=5x-9

The other equation is...

y=5x+19

Let's subtract the second equation from the first...

y=5x-9

-(y=5x+19)

y-y=0

5x-5x=0

-9-19=-28

The result is...

0=0-28

NONSENSE!!!

There is no solution.

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

y=6x+14

12x-2y=-28

Let's place both equations in the same format.

The second equation has the constant isolated.  Let's work with that.  The constant is -28.  Let's note the fact that the x is multiplied by 12 and the y is multiplied by -2.  All three numbers are multiples of 2.  Let's also note the fact that the constant in the first equation is 14, half of positive 28.

12x-2y=-28

(12x-2y)/2=-28/2

6x-y=-14

-1(6x-y)=-1(-14)

y-6x=14

Now, the first equation...

y=6x+14

Let's isolate the 14 by subtracting 6x from both sides...

-6x+y=6x+14-6x

y-6x=14

It results in the same equation.

The solutions to the value of x and y are infinite.