Let's analyze the system of equations:
- \(8x - 2y = -4\)
- \(4x - y = -2\)
First, we'll simplify and rewrite the equations to make them easier to analyze.
Starting with the first equation, we can simplify it by dividing everything by 2: \[ 4x - y = -2 \quad \text{(After dividing by 2)} \]
Now we have:
- \(4x - y = -2\) (from the first equation)
- \(4x - y = -2\) (the second equation is already the same)
When we see that both equations are identical, it indicates that they represent the same line. This means that there are infinitely many solutions since every point on the line is a solution.
Now let's evaluate each statement provided:
-
Solve this system by elimination since it is already in standard form and lined up nicely.
This statement is true, but since the lines are the same, elimination would show infinitely many solutions. -
There is exactly one solution to this system of linear equations and it is (2, 0).
This statement is false; there are infinitely many solutions. -
There are infinite solutions to this system of linear equations.
This statement is true. -
These lines are parallel, so there is no solution to this system of linear equations.
This statement is false; the lines are the same, not parallel and separate. -
There is exactly one solution to this system of linear equations and it is (0, -2).
This statement is false; there are infinitely many solutions. -
Solve this system by substitution since one of the variables is defined by the other without having to do any math.
This statement is not entirely appropriate, as there's no need to use substitution or elimination since the equations are identical.
Final Answers:
- True: “Solve this system by elimination since it is already in standard form and lined up nicely.”
- True: “There are infinite solutions to this system of linear equations.”