How many solutions does the system of equations have? y – 5x = –6 and 3y – 15x = –12 (1 point) Responses one one two two infinitely many infinitely many none
The system of equations can be rewritten as:
1) y - 5x = -6
2) 3y - 15x = -12
By multiplying equation 1 by 3, we get:
3(y - 5x) = 3(-6)
3y - 15x = -18
Comparing this with equation 2, we can see that both equations are equivalent, meaning they represent the same line.
Therefore, the system of equations represents the same line and has infinitely many solutions.
The correct response is: infinitely many.
To determine the number of solutions, we can use the method of elimination or substitution. Let's use the method of elimination.
First, let's multiply the first equation by 3 to make the coefficients of y the same:
3(y - 5x) = 3(-6)
3y - 15x = -18
Now, we can compare this equation to the second equation:
3y - 15x = -18
3y - 15x = -12
The two equations are equivalent, meaning they represent the same line. Thus, the system of equations has infinitely many solutions.
Therefore, the correct answer is "infinitely many."
To determine the number of solutions for a system of equations, we need to examine the relationship between the equations.
Let's analyze the given system of equations:
Equation 1: y – 5x = –6
Equation 2: 3y – 15x = –12
We can solve this system using various methods, such as substitution or elimination. Here, we will use the elimination method.
To eliminate x, we need to multiply Equation 1 by 3:
3(y – 5x) = 3(-6)
3y – 15x = -18
Now, we can compare Equation 2 and the modified Equation 1:
3y – 15x = –12
3y – 15x = –18
As we can see, both equations have the same left-hand side: 3y – 15x.
Now, let's examine the right-hand side of the equations:
-12 ≠ -18
Since the right-hand sides are not equal, we can conclude that the system of equations is inconsistent and has no solutions. Hence, the correct answer is "none."