Are my answers correct?
Tell whether the system is inconsistent OR dependent.
1.) -7x+6y=-4
14x-12y=8
2.) 3x-y=7
-6x+2y=8
My answers:
1.) inconsistent
2.) Dependent
To determine whether a system of equations is inconsistent or dependent, you need to solve the system by combining the equations and analyzing the resulting solution.
For the first system:
1.) -7x+6y=-4
14x-12y=8
To solve this system, you can multiply the first equation by 2 to make the coefficients of x in both equations match:
-14x + 12y = -8
Now, you can add the two equations together:
-14x + 12y + 14x - 12y = -8 + 8
0 = 0
The resulting equation 0 = 0 is always true, which means that the equations are dependent. This means that the two equations represent the same line or multiple lines that intersect at infinitely many points. Therefore, your answer for this system being dependent is correct.
For the second system:
2.) 3x-y=7
-6x+2y=8
One method to solve this system is by multiplying the first equation by 2:
6x - 2y = 14
Now, you can add the two equations together:
6x - 2y - 6x + 2y = 14 + 8
0 = 22
The resulting equation 0 = 22 is always false, which means that the equations are inconsistent. This means that the two equations represent parallel lines that never intersect. Therefore, your answer for this system being inconsistent is correct.
In conclusion, your answers are correct:
1.) The system of equations is dependent.
2.) The system of equations is inconsistent.