How many solutions does the system of equations have? y = 6x + 2 and 3y – 18x = 12 (1 point) Responses one one two two infinitely many infinitely many none
The system of equations y = 6x + 2 and 3y – 18x = 12 can be solved by substitution or elimination.
Let's use substitution:
From the first equation, we can rewrite y = 6x + 2 as y - 6x = 2.
Substitute this expression for y in the second equation:
3(y - 6x) - 18x = 12
3y - 18x - 18x = 12
3y - 36x = 12
Now, let's rearrange this equation to solve for y:
3y = 36x + 12
y = 12x + 4
We have simplified the second equation to y = 12x + 4.
Comparing this with y = 6x + 2, we can see that the two equations have the same slope (or rate of change, which is 6) but different y-intercepts (2 and 4).
Therefore, the system of equations has only one solution.
The correct response is: one
To determine the number of solutions, we can use the method of substitution or elimination. Let's use the substitution method.
Given the system of equations:
1) y = 6x + 2
2) 3y - 18x = 12
We can substitute equation 1) into equation 2) to solve for x:
3(6x + 2) - 18x = 12
18x + 6 - 18x = 12
6 = 12
The equation 6 = 12 is not true, which means the system of equations is inconsistent and has no solutions.
Therefore, the correct response is: "none"
To determine the number of solutions for the system of equations, we can solve them simultaneously and analyze the result.
First, we will start by rewriting the system of equations:
Equation 1: y = 6x + 2 ----(i)
Equation 2: 3y – 18x = 12 ----(ii)
We can now proceed with the solution:
Step 1: Substituting Equation (i) into Equation (ii), we get:
3(6x + 2) - 18x = 12
18x + 6 - 18x = 12
6 = 12
Step 2: Simplifying the equation obtained in Step 1, we find that 6 equals 12, which is not true.
Therefore, the equation 6 = 12 demonstrates a contradiction, indicating that there are no solutions to this system of equations.
Hence, the correct response is "none."