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

Part 1) Infinite solutions

Part 2) No solutions

Part 3) Infinite solutions

Part 4) No solutions

Step-by-step explanation:

Part 1) we have

Group the variables

-------> equation A

-------> equation B

Multiply by equation B

------>

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

-------> equation A

Isolate the variable y

 ------> Divide by both sides

-------> 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

therefore

Line A and Line B are parallel lines

The system has no solution

Part 3) we have

-------> equation A

-------> equation B

Multiply by equation A

------->  

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

Isolate the variable y

-------> equation A

Isolate the variable y

------> Divide by both sides

-------> 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

therefore

Line A and Line B are parallel lines

The system has no solution

For a polygon ABCD, the diagonals are AC and BD.
The midpoint formulas are:
x M = ( x 1 + x 2 )/2    and   y M = ( y 1 + y 2 ) / 2
For AC:
x M = ( x A + x C ) / 2 = ( 1 + ( - 1 ) ) / 2 = 0
y M = ( y A + y C ) / 2 = ( 4 + ( - 4 ) ) / 2 = 0
Therefore: M ( 0, 0 ) - the midpoint of the diagonal AC.
For BD:
x M = ( x B + x D ) / 2  = ( 4 + ( - 4 ) ) / 2  = 0
y M = ( y B + y D ) / 2 = ( - 1 + 1 ) / 2 = 0
M ( 0, 0 ).
Because the midpoint of the both diagonals is the same ( 0, 0 ), we cam say that the diagonals of polygon ABCD bisect each other.