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

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 are infinite solutions 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).

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 start by solving the given system of equations using the elimination method and then check the statements provided.

The system of equations is:

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

To use the elimination method, we can manipulate these equations to eliminate one of the variables.

First, we can multiply the second equation by 2 to help align the coefficients of \( y \):

\[ 2(4x - y) = 2(-2) \implies 8x - 2y = -4 \]

This means the second equation is equivalent to the first equation, which indicates that both equations actually represent the same line.

Since both equations represent the same line, there are infinitely many solutions, as any point that lies on that line is a solution to the system.

Now let's evaluate each statement provided:

1. There is exactly one solution to this system of linear equations and it is (2, 0).
   False. There are infinitely many solutions.

2. There are infinite solutions to this system of linear equations.
   True. Both equations are the same, leading to infinitely many solutions.

3. These lines are parallel, so there is no solution to this system of linear equations.
   False. The lines are not parallel; they are the same line.

4. There is exactly one solution to this system of linear equations and it is (0, -2).
   False. There are infinitely many solutions, and (0, -2) is just one of them.

5. Solve this system by substitution since one of the variables is defined by the other without having to do any math.
   False. While substitution could work, the claim that it can be done without any math is misleading since solving would still involve some algebraic steps.

In summary, the correct statements are:

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