Question

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.

Answer 1

To determine which statements correctly describe the solution to the system of equations given, we can first analyze the system:

1. Equations:

   • \( 8x - 2y = -4 \) (Equation 1)
   • \( 4x - y = -2 \) (Equation 2)

2. Simplifying these equations:

   • Let's simplify Equation 2 by multiplying it by 2 to help see the relationship between the two equations: \[ 2(4x - y) = 2(-2) \implies 8x - 2y = -4 \]
   • This shows that both Equation 1 and Equation 2 are actually the same.

3. Conclusion from simplification:

   • Since both equations are equivalent, they represent the same line. Therefore, there are infinite solutions to this system, as every point on this line is a solution.

Now we can evaluate each response:

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

   • This statement is somewhat misleading; while the equations are in standard form, elimination doesn't apply as they are not independent.

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

   • Incorrect; there are infinitely many solutions.

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

   • Correct; both equations are equivalent.

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

   • Incorrect; the lines are not parallel; they are the same line.

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

   • Incorrect; 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.

   • Misleading; substitution isn't necessary as the lines are already equivalent.

From the analysis, the only correct statement is:

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