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 analyze the system of equations and determine which statements correctly describe its solution, let's first rewrite the given system:

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

We can simplify these equations to check for relationships:

1. Divide the first equation by 2: $$4x-y=-2$$

This is the same as the second equation.

Since both equations represent the same line, they are dependent, which means there are infinitely many solutions to this system of equations (every point on the line defined by the equations is a solution).

Now we evaluate the given statements:

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

   • While elimination is possible, the statement does not clarify that the forms are identical and thus already lead to infinite solutions. This statement is not appropriate.

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

   • This is incorrect because there are infinitely many solutions. False.

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

   • This is correct; we established that both lines are the same. True.

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

   • This is incorrect since the lines are identical, not parallel. False.

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

   • This is also incorrect for the same reason as statement 2; there are infinitely many solutions. False.

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

   • While substitution could work, it overlooks the fact that both equations are identical. Not a strong statement.

In conclusion, the only statement that accurately describes the solution to the system is:

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