x - 2y = 6
3x - 6y = 18
One Solution
No Solution
Infinite Solution
Infinite Solution
To determine whether the system of equations has one solution, no solution, or infinite solutions, we can use methods such as substitution, elimination, or graphing. Let's solve these equations using the elimination method:
Given equations:
1) x - 2y = 6
2) 3x - 6y = 18
Step 1: Multiply equation 1 by 3 to make the coefficients of x in both equations the same:
3 * (x - 2y) = 3 * 6
This simplifies to:
3x - 6y = 18
Step 2: Compare equation 2 with the new equation obtained in step 1:
3x - 6y = 18
3x - 6y = 18
Both equations are identical, which means they represent the same line. Therefore, these equations have infinite solutions.
Hence, the given system of equations has infinite solutions.
To determine whether the given system of equations has one solution, no solution, or infinite solutions, we can use the concept of linear dependence.
First, let's rewrite the equations in the standard form Ax + By = C:
Equation 1: x - 2y = 6
Equation 2: 3x - 6y = 18
We can start by rearranging Equation 1:
x - 2y = 6
⇒ x = 6 + 2y
Now substitute this value of x into Equation 2:
3(6 + 2y) - 6y = 18
⇒ 18 + 6y - 6y = 18
⇒ 18 = 18
As we can see, the equation 18 = 18 holds true. This means that the two equations are dependent, which indicates that they represent the same line. In other words, any solution of Equation 1 will also be a solution of Equation 2.
Since the two equations represent the same line, there are infinitely many solutions. Therefore, the given system of equations has infinite solutions.