How many solutions does the system of equations have?
3x=-12y+15 and x + 4y=5
Y=6x+2 and 3y-18x=12
x-2y=6 and 3x – 6y =18
y-5x=-6 and 3y-15x=-12
Ok, so I’m having problems with these, so if Steve or Reiny could help with these problems and show the work for them that would help me study that would be great, however I need this done in 20 min. ( I know it’s bad having time limits) but unfortunately I need it done soon… so if you can help, awesome.
3x=-12y+15 and x + 4y=5
These two equations are the same. There is a y for any x.
Y=6x+2 and 3y-18x=12
3y = 18 x + 12 is y = 6 x + 4
These are parallel lines that never intersect, no solution
x-2y=6 and 3x – 6y =18
again divide equation 2 by 3 and get
x - 2 y - 6
the two equations are the same, infinite solutions again
y-5x=-6 and 3y-15x=-12
divide second one by 3 and get
y - 5 x = -4
Again two parallel lines never cross, no solutions
Thank you Damon~!
Were they right?
Sure, I can help you with those system of equations problems. Let's go through each one and find the number of solutions it has.
1) 3x = -12y + 15 and x + 4y = 5
To find the number of solutions, we can use the elimination method or substitution method. Let's use the elimination method here:
Multiply the second equation by 3 to make the coefficients of x the same:
3(x + 4y) = 3(5)
3x + 12y = 15
Now, we have two equations:
3x = -12y + 15
3x + 12y = 15
Adding both equations together eliminates x:
3x + 12y + 3x = -12y + 15 + 15
6x + 12y = 30
Dividing both sides of the equation by 6, we get:
x + 2y = 5
This equation is equivalent to the second equation we started with, so these two equations are dependent. It means they represent the same line and have infinitely many solutions.
Therefore, the system of equations has infinite solutions.
2) y = 6x + 2 and 3y - 18x = 12
Let's solve this system using substitution method:
Substitute the value of y from the first equation into the second equation:
3(6x + 2) - 18x = 12
18x + 6 - 18x = 12
6 = 12
This is a false statement, which means these equations are inconsistent and have no solutions.
Therefore, the system of equations has no solutions.
3) x - 2y = 6 and 3x - 6y = 18
Let's solve this system using elimination method:
Multiply the first equation by 3 to make the coefficients of x the same:
3(x - 2y) = 3(6)
3x - 6y = 18
Now, we have two equations:
3x - 6y = 18
3x - 6y = 18
Subtracting the second equation from the first equation will eliminate x and y:
(3x - 6y) - (3x - 6y) = 18 - 18
0 = 0
This is a true statement, which means these equations are dependent. They represent the same line and have infinitely many solutions.
Therefore, the system of equations has infinite solutions.
4) y - 5x = -6 and 3y - 15x = -12
Let's solve this system using elimination method:
Multiply the first equation by 3 to make the coefficients of y the same:
3(y - 5x) = 3(-6)
3y - 15x = -18
Now, we have two equations:
3y - 15x = -18
3y - 15x = -12
Subtracting the second equation from the first equation will eliminate y and x:
(3y - 15x) - (3y - 15x) = -18 - (-12)
0 = -6
This is a false statement, which means these equations are inconsistent and have no solutions.
Therefore, the system of equations has no solutions.
I hope this helps! Let me know if you have any other questions.