Determine Number of Solutions in System

Question

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.

-5x+3y=5

-10x+6y=12

Answer 1

-5x+3y=5

-10x+6y=12

I am think

First, let's rearrange the equations just to make it simpler.

3y = 5x + 5

6y = 10x + 12

We need to eliminate one of our variables, so we will multiply the top equation by 2 so that the y's have the same coefficient.

6y = 10x + 10 and leaving the bottom equation like it is

6y = 10x + 12

Now, subtract the second equation from the first

0y = 0x - 2 which really means 0 = -2

Since 0 can NEVER equal -2, there are no solutions to this system.

You can also work this system by substitution.

Taking the first equation : 3y = 5x + 5

Divide through by 3 y = (5/3) x + 5/3

Now substitute that into the second equation for y

6y = 10x + 12

6[(5/3)x + 5/3] = 10x + 12

Get rid of your parenthesis

(30/3)x + 30/3 = 10x + 12

Simplify your fractions

10x + 10 = 10x + 12

Subtract 10x from both sides and you are left with

10 = 12 never gonna happen so there are no solutions.