How many solutions does the system have? 12x−15y=18 4x−5y=6 (1 point) Responses no solution no solution two solutions two solutions infinitely many solutions infinitely many solutions one solution one solution

Question

How many solutions does the system have? 12x−15y=18 4x−5y=6 (1 point) Responses no solution no solution two solutions two solutions infinitely many solutions infinitely many solutions one solution one solution

Answer 1

The system of equations is given by:

12x − 15y = 18
4x − 5y = 6

We can solve this system by multiplying the second equation by 3 so that the coefficients of x in both equations will be equal:

12x − 15y = 18
12x − 15y = 18

The two equations are identical, so they represent the same line. This means that there are infinitely many solutions to the system.

So the answer is: infinitely many solutions

Answer 2

To determine the number of solutions for this system of equations, you can use the method of solving simultaneous equations, such as substitution or elimination.

Let's use the elimination method to find the number of solutions:

Given system of equations:
12x - 15y = 18 ---(equation 1)
4x - 5y = 6 ---(equation 2)

To eliminate one variable, we'll multiply both sides of equation 2 by 3:
12x - 15y = 18 ---(equation 1)
12x - 15y = 18 ---(equation 2')

Now, we can observe that both equations have identical terms. This means the two equations represent the same line and they are proportional to each other. In this case, there are infinitely many solutions. So the correct response would be "infinitely many solutions."

Answer 3

To determine the number of solutions for the system of equations, we can use the method of elimination or substitution. Let's use the elimination method.

Given system of equations:
12x - 15y = 18 ...(Equation 1)
4x - 5y = 6 ...(Equation 2)

To eliminate one variable, we can multiply Equation 2 by 3:
12x - 15y = 18 ...(Equation 1)
12x - 15y = 18 ...(Equation 2')

Now, if we subtract Equation 2' from Equation 1:
(12x - 15y) - (12x - 15y) = 18 - 18
0 = 0

This means that both equations are equivalent. By graphing these equations, we can see that the two lines representing them overlap and are the same line. Therefore, there are infinitely many solutions for this system of equations.

The correct response is:
infinitely many solutions