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

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

CHOOSE ALL THAT APPLY DUMMY

1 answer

To analyze the system of equations:

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

First, let's manipulate the first equation to see if it can be simplified to compare with the second equation.

The first equation can be simplified by dividing everything by 2:

\[ 4x - y = -2 \]

Now we see that both equations, after simplifying the first, are identical:

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

Since both equations describe the same line, we conclude that the system has infinite solutions.

Based on this analysis, the correct responses are:

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

The options "There is exactly one solution to this system of linear equations and it is (2, 0)," "These lines are parallel, so there is no solution to this system of linear equations," and "There is exactly one solution to this system of linear equations and it is (0, -2)" are incorrect. "Solve this system by substitution since one of the variables is defined by the other without having to do any math" is not relevant, as the simpler method of elimination is appropriate here, but it would be applicable in certain contexts.

Choose these two options:

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