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 given system of equations:

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

Let's first simplify these equations and check their characteristics.

Starting with the second equation, we can express y in terms of x:

From \(4x - y = -2\), we have: \[ y = 4x + 2 \]

Now, substituting \(y\) into the first equation:

\[ 8x - 2(4x + 2) = -4 \] Distributing: \[ 8x - 8x - 4 = -4 \] Which simplifies to: \[ -4 = -4 \] This is a true statement, which means the two equations represent the same line. Hence, there are infinitely many solutions.

Now, reviewing the statements:

1. Correct - Solve this system by elimination since it is already in standard form and lined up nicely. (Not a definitive statement about the solution but somewhat correct in context)
2. Correct - There are infinite solutions to this system of linear equations.
3. Incorrect - These lines are parallel, so there is no solution to this system of linear equations. (They are the same line, not parallel)
4. Incorrect - There is exactly one solution to this system of linear equations and it is (0, -2). (This is incorrect since there are infinite solutions, not just one)
5. Incorrect - There is exactly one solution to this system of linear equations and it is (2, 0). (Again, incorrect due to infinite solutions)
6. Incorrect - Solve this system by substitution since one of the variables is defined by the other without having to do any math. (Not correct; while substitution is valid, we cannot claim it requires no math at all)

So, the statements that apply are:

• Solve this system by elimination since it is already in standard form and lined up nicely.
• There are infinite solutions to this system of linear equations.