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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

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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

[tex]3x=-12y+15[/tex]

Group the variables

[tex]3x+12y=15[/tex] -------> equation A

[tex]x+4y=5[/tex] -------> equation B

Multiply by [tex]3[/tex] equation B

[tex]3*(x+4y)=3*5[/tex] ------> [tex]3x+12y=15[/tex]

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

[tex]y=6x+2[/tex] -------> equation A

[tex]3y-18x=12[/tex]

Isolate the variable y

[tex]3y=18x+12[/tex]  ------> Divide by [tex]3[/tex] both sides

[tex]y=6x+4[/tex] -------> 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 [tex]m=6[/tex]

therefore

Line A and Line B are parallel lines

The system has no solution

Part 3) we have

[tex]x-4y=12[/tex] -------> equation A

[tex]5x-20y=60[/tex] -------> equation B

Multiply by [tex]5[/tex] equation A

[tex]5*(x-4y)=5*12[/tex] ------->[tex]5x-20y=60[/tex]  

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

[tex]y-6x=-3[/tex]

Isolate the variable y

[tex]y=6x-3[/tex] -------> equation A

[tex]4y-24x=-16[/tex]

Isolate the variable y

[tex]4y=24x-16[/tex] ------> Divide by [tex]4[/tex] both sides

[tex]y=6x-4[/tex] -------> 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 [tex]m=6[/tex]

therefore

Line A and Line B are parallel lines

The system has no solution


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