What is the solution of the following system?

Question

Use the elimination method.

{4x+2y=66x−4y=−12

Responses

The solution is (32,  0) .
The solution is begin ordered pair 3 over 2comma 0 end ordered pair .

The solution is (0, 3) .
The solution is begin ordered pair 0 comma 3 end ordered pair .

There are an infinite number of solutions.
There are an infinite number of solutions.

There is no solution.

Answer 1

To solve the system of equations using the elimination method, we start with the given equations:

1. \( 4x + 2y = 6 \)
2. \( x - 4y = -12 \)

Let's simplify the first equation:

1. \( 4x + 2y = 6 \) can be divided by 2:
   \[ 2x + y = 3 \] (Equation 1)

Now we rewrite the second equation: 2. \( x - 4y = -12 \) can be left as is:
\[ x - 4y = -12 \] (Equation 2)

Next, we can express \( y \) in terms of \( x \) using Equation 1:
\[ y = 3 - 2x \]

Now, let's substitute this expression for \( y \) back into Equation 2:

\[ x - 4(3 - 2x) = -12 \]

Expanding and simplifying:

\[ x - 12 + 8x = -12 \] \[ 9x - 12 = -12 \] Adding 12 to both sides: \[ 9x = 0 \] Now, solving for \( x \):

\[ x = 0 \]

Next, we substitute \( x = 0 \) back into the expression for \( y \):

\[ y = 3 - 2(0) = 3 \]

Thus, the solution to the system of equations is:

\[ (x, y) = (0, 3) \]

So, the correct response is:

The solution is (0, 3).