x-2y=6
3x-6y=18 is one solution , no solution , infinite solutions
This is not just one equation, but a system of two equations. We need to solve the system to determine if it has one solution, no solution, or infinite solutions.
Let's solve the system of equations:
Given the equations:
x - 2y = 6 ---(1)
3x - 6y = 18 ---(2)
We can rewrite equation (2):
3x = 6y + 18
Dividing both sides by 3, we get:
x = 2y + 6 ---(3)
Now we have two equations:
x - 2y = 6 ---(1)
x = 2y + 6 ---(3)
We can see that equation (3) is equivalent to equation (1) and thus represents the same line. This means that both equations are actually the same line and represent an infinite number of solutions.
So, the system of equations has infinite solutions.
To determine whether the given system of equations has one solution, no solution, or infinite solutions, we can solve the system using the method of substitution.
1) Solve the first equation, x-2y=6, for x:
x = 6 + 2y
2) Substitute this value of x into the second equation, 3x-6y=18:
3(6 + 2y) - 6y = 18
18 + 6y - 6y = 18
18 = 18
The equation 18 = 18 is always true, regardless of the value of y. This means that the system of equations has an infinite number of solutions. This is because the two equations represent the same line on a graph, and every point on that line is a valid solution to the system.
Therefore, the given system has infinite solutions.
To determine the solution(s) for the given system of equations, we can use the method of substitution or elimination. Let's start by solving the system using the method of substitution.
Given equations:
1) x - 2y = 6
2) 3x - 6y = 18
We can solve equation 1 for x:
x = 6 + 2y
Now, substitute this expression for x in equation 2:
3(6 + 2y) - 6y = 18
Simplify the equation:
18 + 6y - 6y = 18
18 = 18
The equation simplifies to 18 = 18, which is a true statement. This means that the equation is always true, regardless of the value of y.
Since the equation is always true, there are infinitely many solutions to this system of equations.
Therefore, the solution is "infinite solutions".