I need help determining if these systems of equations are dependent, independent or inconsistent.?

Question

I need help determining if these systems of equations are dependent, independent or inconsistent.

These first two systems of equations needed to be solved by the substitution method.

Thanks!

y = 3x + 11
2x + 3y = 0

x = y + 5
2x – 2y = 12

These next two systems of equations needed to be solved by the addition method.

5x – 3y = -20
3x + 2y = 7

2(y – 5) + 4 = 3(x – 6)
3x -2y = 12

Answer 1

if you get a solution x=? and y=? then independent
and consistent
if the equations are multiples of one another then dependent and consistent
if you get a contradiction then inconsistent

1)
y = 3x + 11.........a
2x + 3y = 0.........b
put a in b gives
2x + 3(3x+11) = 0 so 11x=-11
so x = -1 put this in a or b gives
y = 8
so they are independent equations

2)
x = y + 5...........a
2x – 2y = 12.....b
put a in b
2(y+5) - 2y = 12 expanding gives
10 = 12 contradiction so inconsistent

3)
5x – 3y = -20......a
3x + 2y = 7..........b
gives
10x – 6y = -40......2a
9x + 6y = 21..........3b
add 2a + 3b gives
19x = -19
so x = -1 put this in a or b gives
y = 5
so they are independent equations

4)
2(y – 5) + 4 = 3(x – 6)......a
3x -2y = 12 .......................b
expand and rearrange a gives
3x -2y = 12 so a= b
so the equations are dependent means you really only have one equation so there are a whole line of solutions ie infinitely many solutions

the end
.

Answer 2

1. Use the value for y from equation 1 into eqn 2:
2x + 3*(3x + 11) = 0. This gives x = - 3. Using this value of x into eqn 1, find y = 2. This is an independent system.

2. Divide eqn 2 by 2 to get x - y = 6. Then eqn 1 says
x - y = 5. Clearly this is inconsistent.

3. Multiply eqn 1 by 2 and eqn 2 by 3:
10x - 6y = -40
9x + 6y = 21.
Adding gives 19x = -19. Continue, find the system is independent.

4. Clean up eqn 1, multiply eqn 2 by -1 and obtain:
2y - 3x = -12 for both. This is dependent.