How do you solve using the addition method and determine if the system is independant, dependant, inconsistant given 2y-x=3 and x=3y-5?

Question

Abigail Allen · Accepted Answer

I found:
#x=1#
#y=2#
independent

We can add together, in columns, the two equations after a bit of rearranging to make them easier to add: #{(-x+2y=3),(x-3y=-5):}# #color(red)(0-y=-2# So that #y=2# substituting back into, say, your first equation we get: #-x+4=3# and: #x=1# One solution: system consistem and independent.

Isaiah Anthony · Answer

undefined To solve the system of equations using the addition method:

1. Rearrange the equations to standard form: 
   - Equation 1: 2y - x = 3
   - Equation 2: x = 3y - 5
   
2. Substitute the expression for x from Equation 2 into Equation 1:
   2y - (3y - 5) = 3
   
3. Simplify the equation:
   2y - 3y + 5 = 3
   -y + 5 = 3
   
4. Move constants to one side:
   -y = 3 - 5
   
5. Simplify:
   -y = -2
   
6. Multiply both sides by -1 to isolate y:
   y = 2
   
7. Substitute the value of y into Equation 2 to find the value of x:
   x = 3(2) - 5
   x = 6 - 5
   x = 1
   
Now, we have found that y = 2 and x = 1. To determine if the system is independent, dependent, or inconsistent, we need to check if the solution satisfies both equations:

- Substitute the values of x and y into both equations:
   Equation 1: 2(2) - 1 = 3  (True)
   Equation 2: 1 = 3(2) - 5    (True)
   
Since the solution (x=1, y=2) satisfies both equations, the system is independent and consistent.

Ethan Adkins · Answer

undefined To solve the system of equations using the addition method, first, rewrite the equations in standard form if they are not already in that form. Then, add or subtract the equations to eliminate one of the variables. Solve for the remaining variable, and then substitute that value back into one of the original equations to find the value of the other variable. To determine if the system is independent, dependent, or inconsistent, analyze the resulting solution. If you obtain a unique solution for both variables, the system is independent. If the equations simplify to a true statement (such as 0 = 0) when solving, the system is dependent. If the equations simplify to a false statement (such as 0 = 1) when solving, the system is inconsistent.