3x+2y = 0

x-2y = 0
If the system has infinitely many solutions, express your answer in the form x=x and y as a function of x
Your answer is
x=? y=?

Add the two equations together to get

4x=0, therefore x=0.
Substitute x=0 into the first equation to get y=0 also.
Thus x=0, y=0.

To solve the given system of equations:

Equation 1: 3x + 2y = 0
Equation 2: x - 2y = 0

We can solve this system of equations by using the method of elimination:

Step 1: Multiply equation 2 by 3.
3(x - 2y) = 3(0)
This simplifies to:
3x - 6y = 0

Step 2: Add equation 1 and the modified equation 2 (from step 1).
3x + 2y + 3x - 6y = 0 + 0
Combining like terms:
6x - 4y = 0

Step 3: Simplify the equation obtained in step 2.
Divide the entire equation by 2:
3x - 2y = 0

Step 4: Now we have two equations: 3x + 2y = 0 and 3x - 2y = 0.
We can see that both equations are exactly the same, just with a different sign for the y-term.

Step 5: Therefore, this system has infinitely many solutions.
We express the solution in the form x = x and y as a function of x:
x = x (Where x can take any real value)
y = -3/2x

So, the answer is x = x and y = -3/2x.

To determine if the given system of equations has infinitely many solutions, we can try to solve the system and look for any patterns or relationships between the variables.

Let's solve the system of equations:

1. Start with the given system:
3x + 2y = 0 ...(Equation 1)
x - 2y = 0 ...(Equation 2)

2. Multiply Equation 2 by 3 to create a cancellation opportunity for the y variable:
3(x - 2y) = 3(0)
3x - 6y = 0 ...(Equation 3)

3. Now, we have two equations with the same value for 3x. Subtract Equation 3 from Equation 1:
(3x + 2y) - (3x - 6y) = 0 - 0
3x + 2y - 3x + 6y = 0
8y = 0

4. Simplify Equation 4:
8y = 0
Divide both sides by 8:
y = 0

5. Substitute the value of y back into Equation 2 to solve for x:
x - 2(0) = 0
x - 0 = 0
x = 0

6. Therefore, the solution to the system of equations is:
x = 0
y = 0

Since we have a unique solution (x=0, y=0), the system does not have infinitely many solutions.