Solve the system

6x+4y = 12
9x+6y = 18
If the system has infinitely many solutions, express your answer in the form x=x and y as a function of x
x=? y=?

did you notice that if you divide the first equation by 2, and the second equation by 3, you end with the same equation 3x + 2y = 6 ??

So really you are given only one equation, which of course has an infinite number of solutions.

now re-arrange 3x + 2y = 6 to
2y = -3x + 6
y = (-3/2)x + 3

you can re-arrange for x in the same way.

6x + 9y = –18

6x + 4y = 12

To solve the system of equations:

Step 1: Multiply the first equation by 3 and the second equation by 2 to make the coefficients of x in both equations the same.
3(6x+4y) = 3(12)
2(9x+6y) = 2(18)

Simplifying these equations, we get:
18x + 12y = 36
18x + 12y = 36

Step 2: We notice that the two equations are identical, meaning they represent the same line. This means that the system has infinitely many solutions.

Step 3: To express the solution in the form x=x and y as a function of x, we can choose any value for x and solve for y.

Let's choose x=0:
18(0) + 12y = 36
12y = 36
y = 36/12
y = 3

So, the solution is x = x and y = 3.

To solve this system of linear equations, we can use the method of elimination. The goal is to eliminate one variable by manipulating the equations so that when we add or subtract them, one variable cancels out.

Let's start by multiplying the first equation by 3 and the second equation by 2, so that the coefficients of x in both equations will be equal:

3(6x + 4y) = 3(12) => 18x + 12y = 36
2(9x + 6y) = 2(18) => 18x + 12y = 36

Now, observe that both of the resulting equations are equal. This means that the two equations in the system are dependent and represent the same line. Therefore, the system has infinitely many solutions.

When a system has infinitely many solutions, it means that all points on the line are solutions to the system. So, the solution can be expressed as x = x (where x can be any real number) and y as a function of x.

In this case, we can express y in terms of x by rearranging one of the equations. Let's rearrange the first equation:

6x + 4y = 12
4y = 12 - 6x
y = (12 - 6x)/4

Therefore, the solution is x = x and y = (12 - 6x)/4.