Choose ALL that apply. Select the statement(s) that correctly describes the solution to this system of equations
8x−2y=−4
4x−y=−2
1 Solve this system by elimination since it is already in standard form and lined up nicely.
2 There is exactly one solution to this system of linear equations and it is (2, 0).
3 There are infinite solutions to this system of linear equations.
4 These lines are parallel, so there is no solution to this system of linear equations.
5 There is exactly one solution to this system of linear equations and it is (0, -2).
6 Solve this system by substitution since one of the variables is defined by the other without having to do any math.
The correct statements are:
2 There is exactly one solution to this system of linear equations and it is (2, 0).
6 Solve this system by substitution since one of the variables is defined by the other without having to do any math.
To determine the correct statements that describe the solution to the system of equations:
1. Solve this system using elimination: This statement is not accurate. Although the system is already in standard form, it is not lined up nicely for elimination.
2. There is exactly one solution to this system of linear equations and it is (2, 0): This statement is not correct. The system of equations does not have this specific solution.
3. There are infinite solutions to this system of linear equations: This statement is not true. The given system does not have an infinite number of solutions.
4. These lines are parallel, so there is no solution to this system of linear equations: This statement is accurate. The given system represents two parallel lines, which means there is no solution.
5. There is exactly one solution to this system of linear equations and it is (0, -2): This statement is not correct. The system of equations does not have this specific solution.
6. 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 accurate. The system does not allow for direct substitution without performing any calculations.
Therefore, the correct statement from the given options is:
4. These lines are parallel, so there is no solution to this system of linear equations.
To determine the correct statement(s) that describe the solution to the system of equations, let's go through the given options one by one.
1. Solve this system by elimination since it is already in standard form and lined up nicely.
This statement suggests using the elimination method, which involves eliminating one variable by adding or subtracting the equations. However, to use this method, the coefficients of either x or y in the two equations need to be the same or multiples of each other. In this case, the coefficients of x in the two equations (8 and 4) are not multiples of each other, so using elimination as described is not suitable. Therefore, this statement is not correct.
2. 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 it:
8x - 2y = -4 ...(1)
4x - y = -2 ...(2)
We can multiply equation (2) by 2 to make the coefficients of y in both equations the same:
8x - 2y = -4 ...(1)
8x - 2y = -4 ...(3)
Now we can see that equations (1) and (3) are equivalents. These equations represent the same line, and the solution to the system is all the points on that line. This means that there are infinitely many solutions to this system. Hence, statement 2 is not correct.
3. There are infinite solutions to this system of linear equations.
As explained earlier, this statement is correct because the system of equations represents the same line, and all the points on that line are solutions to the system.
4. These lines are parallel, so there is no solution to this system of linear equations.
This statement is not correct since the two equations represent the same line, not parallel lines. Parallel lines would have different slopes, which is not the case here.
5. There is exactly one solution to this system of linear equations, and it is (0, -2).
This statement is not correct as we found that there are infinite solutions, not just one.
6. Solve this system by substitution since one of the variables is defined by the other without having to do any math.
Using substitution involves solving one equation for one variable and then substituting that expression into the other equation. In this case, neither equation defines one variable in terms of the other, so this statement is not correct.
Therefore, the correct statement for the solution to this system of equations is:
3. There are infinite solutions to this system of linear equations.