prelanatouri
Answered

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

Verified Answer :

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

calculista

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


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