For 2x-5y=16, 6x-15y=30 I think there are zero number of solutions.
I used the elimination method and got 0=0. Does that mean zero number of solutions?
Actually you made an error:
2x-5y=16, 6x-15y=30
multiply the first by 3 ---->
6x - 15y = 48
subtract the second
0 = 18
If your variable drops out and you end up with a true statement, such as your 8=8, then you have an infinite number of solutions. You are trying to intersect the line with itself.
If your variable drops out and you end up with a false statement, such as 0=5, then there is no solution , you would have two parallel lines
To determine the number of solutions for a system of linear equations, we can solve the equations using various methods such as substitution, elimination, or matrix operations. Let's use the method of elimination to solve the given system.
Given equations:
1) 2x - 5y = 16
2) 6x - 15y = 30
To eliminate one variable, we need to make the coefficients of either x or y the same in both equations. In this case, we'll eliminate y.
Let's start by multiplying equation 1) by 3 to make the coefficient of y the same in both equations. We get:
3 * (2x - 5y) = 3 * 16
which simplifies to:
6x - 15y = 48
Now, we have the following system of equations:
3) 6x - 15y = 48
4) 6x - 15y = 30
By comparing equations 3) and 4), we can see that both equations are identical. There is no difference in their coefficients or constants. Since these equations represent the same line, it means that the system is consistent and dependent. In other words, there are infinitely many solutions.
So, the statement that there are zero solutions is incorrect. The correct answer is that there are infinitely many solutions for the given system of equations.