Question

Select the statement that correctly describes the solution to this system of equations.
8x-2y=-4
4x-y=-2

Responses

There is exactly one solution at (0,–2).
There is exactly one solution at (0,–2).


There is no solution.
There is no solution.


There is exactly one solution at (2,0).
There is exactly one solution at (2,0).

There are infinitely many solutions.

Answer 1

To determine the solution to the system of equations:

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

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

\[
4x - y = -2 \implies y = 4x + 2
\]

Now, substitute \(y\) from this equation into the first equation:

\[
8x - 2(4x + 2) = -4
\]

Expanding this gives:

\[
8x - 8x - 4 = -4
\]

This simplifies to:

\[
-4 = -4
\]

This is a true statement, indicating that the two equations are dependent; they represent the same line. Thus, there are infinitely many solutions along this line.

Therefore, the correct statement is:

**There are infinitely many solutions.**