Question

How many solutions does the system of equations have 3x = -12y + 15 and x + 4y = 5

how many solutions does the system of equations have y = 6x + 2 and 3y - 18x = 12

how many solutions does the system of equations havex - 4y = 12 and 5x - 20y = 60

how many solutions does the system of equations have y - 6x = -3 and 4y - 24x = - 16

Answer 1

Part 1) Infinite solutions

Part 2) No solutions

Part 3) Infinite solutions

Part 4) No solutions

Explanation:

Part 1) we have

`3x=-12y+15`

Group the variables

`3x+12y=15` -------> equation A

`x+4y=5` -------> equation B

Multiply by
`3` equation B

`3*(x+4y)=3*5` ------>
`3x+12y=15`

The equation A and the equation B are the same equation, is the same line

therefore

The system has infinite solutions

Part 2) we have

`y=6x+2` -------> equation A

`3y-18x=12`

Isolate the variable y

`3y=18x+12` ------> Divide by
`3` both sides

`y=6x+4` -------> equation B

we know that

If two lines has the same slope , then they are parallel lines

In this problem the line of the equation A and the line of the equation B has the same slope
`m=6`

therefore

Line A and Line B are parallel lines

The system has no solution

Part 3) we have

`x-4y=12` -------> equation A

`5x-20y=60` -------> equation B

Multiply by
`5` equation A

`5*(x-4y)=5*12` ------->
`5x-20y=60`

The equation A and the equation B are the same equation, is the same line

therefore

The system has infinite solutions

Part 4) we have

`y-6x=-3`

Isolate the variable y

`y=6x-3` -------> equation A

`4y-24x=-16`

Isolate the variable y

`4y=24x-16` ------> Divide by
`4` both sides

`y=6x-4` -------> equation B

we know that

If two lines has the same slope , then they are parallel lines

In this problem the line of the equation A and the line of the equation B has the same slope
`m=6`

therefore

Line A and Line B are parallel lines

The system has no solution

Answer 2

Below are the answers:

1. 3x = - 12y + 15 3x + 12y = 15 reduces to x + 4y = 5 this matches the other equation, therefore, they are on the same line and have INFINITE SOLUTIONS
2. y = 6x + 2 -6x + y = 2 6x - y = -2 3y - 18x = 12 -18x + 3y = 12 18x - 3y = -12 reduces to 6x - y = -4 The equations do not match, therefore, there is NO SOLUTION
3. x - 4y = 12 5x - 20y = 60 reduces to x - 4y = 12 same equation, same line, INFINITE SOLUTIONS
4. y - 6x = -3 -6x + y = -3 6x - y = 3
4y - 24x = -16 -24x + 4y = -16 24x - 4y = 16 reduces to 6x - y = 4
NO SOLUTIONS