Simultaneous Equations: Solving [eq2] and Plugging into [eq1]

[camino]

Homework Statement

[eq1]: (Va-Vb)/(2i) = {[Vb-(10<0)]/(4-8i)} + {(Vb)/(6i)}

[eq2]: (Va-Vb)/(2i) = (8<20) - Va

Homework Equations

Note: < is angle

The Attempt at a Solution

Solving [eq2]

(Va-Vb) = 2i[(8<20) - Va]
(Va-Vb) = (2i)(8<20) - (2i)(Va)
-Vb = (2i)(8<20) - (2i)(Va) - Va
Vb = -(2i)(8<20) + (2i)(Va) + Va
Vb = -(16<110) + (2i)(Va) + Va

Then plugging into [eq1]

[Va + (16<110) + (2i)(Va) + Va]/[2i] = {[-(16<110) + (2i)(Va) + Va - (10<0)]/[4-8i]} + {[-(16<110) + (2i)(Va) + Va]/[6i]}

[2Va + (16<110) + (2i)(Va)]/[2i] = {[-(16<110) + (2i)(Va) + Va - (10<0)]/[4-8i]} + {[-(16<110) + (2i)(Va) + Va]/[6i]}

I am stuck at this point. If I could find the LCM of the the 2 fractions on the right and add them together I would be fine, but since there are imaginary numbers I'm not sure if that's even possible. (I tried using the lcm feature on my TI-89, didn't work).

Or if there's a better way to solve this please let me know. Thanks.

 

[berkeman]

Homework Statement

[eq1]: (Va-Vb)/(2i) = {[Vb-(10<0)]/(4-8i)} + {(Vb)/(6i)}

[eq2]: (Va-Vb)/(2i) = (8<20) - Va

Homework Equations

Note: < is angle

The Attempt at a Solution

Solving [eq2]

(Va-Vb) = 2i[(8<20) - Va]
(Va-Vb) = (2i)(8<20) - (2i)(Va)
-Vb = (2i)(8<20) - (2i)(Va) - Va
Vb = -(2i)(8<20) + (2i)(Va) + Va
Vb = -(16<110) + (2i)(Va) + Va

Then plugging into [eq1]

[Va + (16<110) + (2i)(Va) + Va]/[2i] = {[-(16<110) + (2i)(Va) + Va - (10<0)]/[4-8i]} + {[-(16<110) + (2i)(Va) + Va]/[6i]}

[2Va + (16<110) + (2i)(Va)]/[2i] = {[-(16<110) + (2i)(Va) + Va - (10<0)]/[4-8i]} + {[-(16<110) + (2i)(Va) + Va]/[6i]}

I am stuck at this point. If I could find the LCM of the the 2 fractions on the right and add them together I would be fine, but since there are imaginary numbers I'm not sure if that's even possible. (I tried using the lcm feature on my TI-89, didn't work).

Or if there's a better way to solve this please let me know. Thanks.

Perhaps it can be solved in that polar format for the complex numbers, but I would find it easier to solve it in rectangular format. That is, convert each complex number from "magnitude>angle" format into A+jB format. Then to solve, the real parts have to be equal, and the imaginary parts have to be equal. You can convert it back to polar form in the end, if that's the format that the complex number answer is supposed to be in.

 

[Architeuthis Dux]

There are a few different approaches you could take to solve this problem. One method would be to first simplify the fractions on the right side of [eq1] by finding the common denominator and then combining them. This would give you a single fraction on the right side, which you could then equate to the left side of the equation and solve for Va. Once you have found the value of Va, you can plug it back into [eq2] to solve for Vb.

Another approach would be to use substitution. From [eq2], you can solve for Va in terms of Vb, and then substitute that expression into [eq1]. This would give you a single equation with one variable, which you can then solve for Vb. Once you have found the value of Vb, you can plug it back into the expression for Va to find its value.

In either case, it may be helpful to convert the polar form of the complex numbers to rectangular form to simplify the calculations. Additionally, it may be helpful to use a calculator or software program to perform the calculations.

Overall, the key to solving simultaneous equations is to manipulate the equations to eliminate one variable and then solve for the remaining variable. Keep in mind that there may be multiple ways to approach a problem, so if one method is not working for you, try another.