1)
If you have at least one solution, it is consistent.
If you have no solutions, it is inconsistent.
When you solve the system of equations, you will get x = -2/3 and y = 3.5. This shows us that the system of equations is independent.
If the system of equations was dependent, you would have gotten a final result like 0 = 0.
Your final answer is C) Consistent and independent.
2)
Coincident lines are lines that look exactly the same when graphed.
You can see this if you divide the second equation by 2.
8y + 2x = 4
Divide both sides by 2.
4y + x = 2
This is exactly like the first equation.
Your final answer is A) Coincident.
3)
We have coincident lines, again.
Coincident lines will have infinitely many solutions because any solution to one equation will also work in the second equation.
Your final answer is D) Infinitely many.
4)
We are dealing with coincident lines, again.
Both equations have to be simplified down to x + 3y = 5.
Obviously, we can see that the second equation is the first equation multiplied by 4. We can tell that because the first equation has x, and the second equation has 4x.
To find the remaining terms, multiply by 4.
x + 3y = 5
Multiply both sides by 4.
4x + 4(3y) = 4(5)
4x + 12y = 20
The missing values are, respectively, 12y and 20.
5)
Graph each line and use the information I have told you above to classify each system.
1) A) Consistent and independent.
2) B) Coincident.
3) C) Inconsistent.
4) C) Inconsistent.
