What is the solution of the following system?

Question

Use the substitution method.

{y−2x=816+4x=2y

Responses

The only solution is (24, 0) .
The only solution is begin ordered pair 24 comma 0 end ordered pair .

The only solution is (1, 10) .
The only solution is begin ordered pair 1 comma 10 end ordered pair .

There is no solution.
There is no solution.

There are an infinite number of solutions.

Answer 1

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

1. \(y - 2x = 8\)
2. \(16 + 4x = 2y\)

Let's solve the first equation for \(y\):

From equation (1): \[ y = 2x + 8 \]

Now, substitute \(y\) in the second equation:

Substituting \(y\) into equation (2): \[ 16 + 4x = 2(2x + 8) \] Now simplify: \[ 16 + 4x = 4x + 16 \] If we rearrange this, we can see that both sides are equal: \[ 16 + 4x - 4x = 16 \] \[ 16 = 16 \] This is a true statement, which means that the two equations are the same line.

Thus, there are infinite solutions to the system of equations as the equations represent the same line.

Therefore, the correct response is:

There are an infinite number of solutions.