Check all that apply.
8x - 2y = -4
4x - y = -2
A: solve this system by elimination since it is already In standard form and lined up nicely
B: there is exactly one solution to this system of linear equations and it is (2,0)
C: there are infinite solutions to this system of linear equations
D: these lines are parallel, so there is no solution to this system of linear equations
E: there is exactly one solution to this system of linear equations and it is (0,-2)
F: solve this system by substitution since one of the variables is defined by the other without have to do any math
B: there is exactly one solution to this system of linear equations and it is (2,0)
To check all that apply, let's solve the system of equations:
8x - 2y = -4 .............(1)
4x - y = -2 .............(2)
By analyzing the coefficients, we can tell that option A is incorrect as elimination is not required here.
To solve the system, we will use substitution.
Let's rearrange equation (2) to solve for y:
y = 4x + 2
Now, substitute this expression for y in equation (1):
8x - 2(4x + 2) = -4
8x - 8x - 4 = -4
-4 = -4
The equation simplifies to -4 = -4, which is true.
Since the equation is true, the system has infinite solutions. Therefore, option C is correct.
Since there are infinite solutions, options B, D, and E are incorrect.
Therefore, the correct options are:
C: there are infinite solutions to this system of linear equations
F: solve this system by substitution since one of the variables is defined by the other without having to do any math.
To solve this system of linear equations, we can use either the elimination or substitution method. Let's go through the options one by one.
A: Solve this system by elimination since it is already in standard form and lined up nicely.
To solve the system by elimination, we need to manipulate the equations so that the coefficients of either x or y are the same in both equations. Looking at the system given:
8x - 2y = -4
4x - y = -2
If we multiply the second equation by 2, we can make the coefficients of y in both equations the same. The modified system becomes:
8x - 2y = -4
8x - 2y = -4
Now, we subtract the second equation from the first equation:
(8x - 2y) - (8x - 2y) = (-4) - (-4)
This simplifies to:
0 = 0
Since 0 = 0, this means that the two equations are equivalent and describe the same line. Therefore, there are infinite solutions to this system of linear equations. Answer choice C is correct.
B: There is exactly one solution to this system of linear equations and it is (2,0).
Since option B claims that there is exactly one solution, it is incorrect. We have determined that there are infinite solutions, not just one.
D: These lines are parallel, so there is no solution to this system of linear equations.
Since option D claims that the lines are parallel and there is no solution, it is also incorrect. The two equations describe the same line, so they are not parallel.
E: There is exactly one solution to this system of linear equations and it is (0,-2).
Since option E claims that there is exactly one solution, it is incorrect. We have determined that there are infinite solutions, not just one.
F: Solve this system by substitution since one of the variables is defined by the other without having to do any math.
To use the substitution method, we need to express one variable in terms of the other in one of the equations. Looking at the given system:
8x - 2y = -4
4x - y = -2
We can solve the second equation for y in terms of x:
y = 4x + 2
Now we substitute this expression for y into the first equation:
8x - 2(4x + 2) = -4
Simplifying this equation gives:
8x - 8x - 4 = -4
-4 = -4
This equation is true, which means that any value of x will satisfy the equation. Therefore, there are infinite solutions to this system of linear equations. Answer choice C is correct.
In conclusion, the correct answers are options C: there are infinite solutions to this system of linear equations, and options A and F are incorrect as they do not align with the method used to solve the system.