Two systems of equations are shown:
System A System B
6x + y = 2 2x − 3y = −10
−x − y = −3 −x − y = −3
Which of the following statements is correct about the two systems of equations?
The value of x for System B will be 4 less than the value of x for System A because the coefficient of x in the first equation of System B is 4 less than the coefficient of x in the first equation of System A.
They will have the same solution because the first equations of both the systems have the same graph.
They will have the same solution because the first equation of System B is obtained by adding the first equation of System A to 4 times the second equation of System A.
The value of x for System A will be equal to the value of y for System B because the first equation of System B is obtained by adding −4 to the first equation of System A and the second equations are identical.
I suggest solving each system, then see which of those complicated-sounding reasons will fit our solution.
I will do the first system, you do the second
6x + y = 2
−x − y = −3 or x + y = 3 ---> y = 3-x
sub that into the other,
6x + (3-x) = 2
6x + 3 - x = 2
5x = -1
x = -1/5
sub that into y = 3-x
y = 3 -(-1/5) =16/5
To determine which statement is correct about the two systems of equations, we can analyze the equations and compare their coefficients. Let's examine each statement:
1. The value of x for System B will be 4 less than the value of x for System A because the coefficient of x in the first equation of System B is 4 less than the coefficient of x in the first equation of System A.
To check this statement, we need to compare the coefficients of x in both systems. In System A, the coefficient of x is 6, and in System B, it is 2. There is no difference of 4 between these coefficients, so this statement is incorrect.
2. They will have the same solution because the first equations of both the systems have the same graph.
This statement suggests that the first equations of both systems represent the same line. To check this, we can rewrite the second equation of System B in the form of the first equation by moving terms around: 2x - 3y = -10 becomes 2x - 3y + 10 = 0. If we compare this equation with the first equation in System A, which is 6x + y = 2, we can see that the coefficients and constants are different. Since the equations represent different lines, this statement is incorrect.
3. They will have the same solution because the first equation of System B is obtained by adding the first equation of System A to 4 times the second equation of System A.
To test this statement, let's multiply the second equation of System A by 4: -4x - 4y = -12. If we add this equation to the first equation of System A, we obtain the first equation of System B: 6x + y + (-4x - 4y) = 2 + (-12) simplifies to 2x - 3y = -10. Since the first equation of System B is indeed obtained by adding the first equation of System A to 4 times the second equation of System A, this statement is correct.
4. The value of x for System A will be equal to the value of y for System B because the first equation of System B is obtained by adding -4 to the first equation of System A, and the second equations are identical.
This statement suggests that x in System A is equal to y in System B. However, based on the equations given, there is no indication of this relationship between x and y. Therefore, this statement is incorrect.
In conclusion, the correct statement about the two systems of equations is: They will have the same solution because the first equation of System B is obtained by adding the first equation of System A to 4 times the second equation of System A.
The correct statement about the two systems of equations is:
They will have the same solution because the first equations of both the systems have the same graph.