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

Question

8x−2y=−4
8
𝑥
−
2
𝑦
=
−
4

4x−y=−2
4
𝑥
−
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

Let's analyze the given equations:

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

First, let's simplify both equations.

The first equation can be simplified by dividing every term by 2: \[ 4x - y = -2 \]

Notice that the second equation is identical to the simplified first equation: \[ 4x - y = -2 \]

Since both equations represent the same line, this implies that there are infinite solutions to the system of equations. Therefore, the parallel lines share all points in common.

Now, let's review the provided statements:

1. Solve this system by elimination since it is already in standard form and lined up nicely. - This statement is technically correct, but elimination may not be the best method here as the two equations are effectively the same line.

2. There is exactly one solution to this system of linear equations and it is (2, 0). - Incorrect, as there are infinitely many solutions.

3. There are infinite solutions to this system of linear equations. - Correct, as both equations represent the same line.

4. These lines are parallel, so there is no solution to this system of linear equations. - Incorrect, as the lines are not just parallel; they are the same line.

5. There is exactly one solution to this system of linear equations and it is (0, -2). - Incorrect, since 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 statement is misleading. While substitution could work, it isn’t necessary for finding the infinite solutions, as both equations are equivalent.

Based on this analysis, the only correct statement is:

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