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.
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 the correct statements about the system of equations:

1. 8x - 2y = -4
2. 4x - y = -2

We can rewrite the first equation in standard form to see if it is equivalent to the second equation.

First, let's divide the first equation by 2 to simplify it:

• \( 8x - 2y = -4 \Rightarrow 4x - y = -2 \)

Now we see that both equations are equivalent:

• 4x - y = -2 (which is the second equation)

Since both equations represent the same line, that means there are infinitely many solutions (any point on the line).

Now let’s evaluate the statements:

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

   • Incorrect: While it can be solved via elimination, the system actually leads to the same equation.

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

   • Incorrect: There are infinitely many solutions, not just one.

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

   • Correct: Since both equations are equivalent, there are infinitely many solutions.

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

   • Incorrect: The lines represent the same line, so they are not parallel.

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

   • Incorrect: Again, 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.

   • Incorrect: Both equations are equivalent and do not need substitution or any further simplification.

Therefore, the only correct statement is:

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