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
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 having to do any math.
c There are infinite solutions to this system of linear equations.
To determine the correct statement(s) that describe the solution to the given system of equations, let's solve the equations:
We are given the system of equations:
1) 8x - 2y = -4
2) 4x - y = -2
We can solve this system using elimination or substitution method.
Using the elimination method:
Multiply equation (2) by 2:
4(4x - y) = 4(-2)
8x - 2y = -4
We can see that equation (2) is equivalent to equation (1). This means that the two equations represent the same line and, therefore, have the same infinitely many solutions.
Hence, the correct statement(s) that describe the solution to this system of equations are:
c) There are infinite solutions to this system of linear equations.
To determine the correct statement(s) that describe the solution to this system of equations, we can apply different methods of solving linear systems. Let's go through each statement and determine if it is correct or not based on the given equations:
a) Solve this system by elimination since it is already in standard form and lined up nicely.
This statement suggests using the elimination method to solve the system. While the equations are in standard form, they are not lined up in a way that facilitates easy elimination. Therefore, this statement is not correct.
b) There is exactly one solution to this system of linear equations, and it is (2, 0).
To determine if this statement is correct, we need to solve the system of equations. Let's solve the given system using any method.
We have the system:
8x - 2y = -4 ...(Equation 1)
4x - y = -2 ...(Equation 2)
We can use the substitution method to solve this system:
From Equation 2, we have y = 4x + 2.
Substituting this into Equation 1:
8x - 2(4x + 2) = -4
Simplifying:
8x - 8x - 4 = -4
-4 = -4
The equation simplifies to -4 = -4, which means the two equations are equivalent. This indicates that the equations represent the same line. Therefore, there are infinitely many solutions, and the statement b is incorrect.
c) There are infinite solutions to this system of linear equations.
As explained above, when we simplify the equations, we find that they represent the same line. Hence, there are infinite solutions. Thus, this statement is correct.
d) These lines are parallel, so there is no solution to this system of linear equations.
Since the equations represent the same line, they are not parallel. Therefore, this statement is incorrect.
e) There is exactly one solution to this system of linear equations, and it is (0, -2).
As established earlier, the system has infinitely many solutions and not just one. Therefore, this statement is incorrect.
f) Solve this system by substitution since one of the variables is defined by the other without having to do any math.
In this case, substituting one variable into the other does involve some calculations. Therefore, this statement is not correct.
To summarize, the correct statement(s) that describes the solution to this system of equations are:
- There are infinite solutions to this system of linear equations (Statement c).