Question

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

Answer 1

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

Answer 2

Answer:

Part 1) Infinite solutions

Part 2) No solutions

Part 3) Infinite solutions

Part 4) No solutions

Step-by-step 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