Check all that apply. Select the statement(s) that correctly describes the solution to this system of equations:

Question

8x−2y=−4

4x−y=−2
(2 points)
Responses

Solve this system by elimination since it is already in standard form and lined up nicely.
Solve this system by elimination since it is already in standard form and lined up nicely.

There is exactly one solution to this system of linear equations and it is (2, 0).
There is exactly one solution to this system of linear equations and it is (2, 0).

There are infinite solutions to this system of linear equations.
There are infinite solutions to this system of linear equations.

These lines are parallel, so there is no solution to this system of linear equations.
These lines are parallel, so there is no solution to this system of linear equations.

There is exactly one solution to this system of linear equations and it is (0, -2).
There is exactly one solution to this system of linear equations and it is (0, -2).

Solve this system by substitution since one of the variables is defined by the other without having to do any math.

Answer 1

To determine which statements about the given system of equations are correct, we need to analyze the system:

1. The equations are: \[ 8x - 2y = -4 \quad (1) \] \[ 4x - y = -2 \quad (2) \]

2. We can simplify the first equation. If we divide the entire equation (1) by 2, we get: \[ 4x - y = -2 \quad (3) \] This is the same as equation (2).

From this simplification, we see that equations (2) and (3) are identical, meaning they represent the same line. Therefore, the situation is as follows:

• There are infinitely many solutions because both equations represent the same line.

Now, we can evaluate each of the statements given:

1. Solve this system by elimination since it is already in standard form and lined up nicely.

   • This is correct in a general sense, but more relevant is the fact they are identical, thus leading to infinite solutions, making it unnecessary.

2. There is exactly one solution to this system of linear equations and it is (2, 0).

   • This is incorrect. There are infinitely many solutions.

3. There are infinite solutions to this system of linear equations.

   • This is correct. As established, both equations are the same.

4. These lines are parallel, so there is no solution to this system of linear equations.

   • This is incorrect. The lines are the same, not parallel.

5. There is exactly one solution to this system of linear equations and it is (0, -2).

   • This is incorrect. There are infinitely many solutions.

6. Solve this system by substitution since one of the variables is defined by the other without having to do any math.

   • This can be considered somewhat ambiguous, but since both equations are identical, solving by substitution would not require additional manipulation. However, it is not the typical approach due to the nature of the results.

Based on the analysis, the correct responses are:

• There are infinite solutions to this system of linear equations.