Learn how to solve for the real variables x and y in equations involving complex numbers! This video covers 4 examples, showing you step-by-step how to equate the real and imaginary parts and solve the resulting system of equations. Perfect for complex number beginners!
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by Jagdish Chawla
"Statistics is the grammar of science."
Karl Pearson
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𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧: Find the value of x and y for (x + y) + (2x - 3y)i = 4i - 3
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:
We are given the equation (x + y) + (2x - 3y)i = -3 + 4i.
Equate the real and imaginary parts of both sides of the equation.
The real part of the left-hand side is (x + y).
The real part of the right-hand side is -3.
Equating the real parts, we get the equation:
x + y = -3 (Equation 1)
The imaginary part of the left-hand side is (2x - 3y).
The imaginary part of the right-hand side is 4.
Equating the imaginary parts, we get the equation:
2x - 3y = 4 (Equation 2)
Now we have a system of two linear equations with two variables. We can solve this using elimination.
Multiply Equation 1 by 3:
3(x + y) = 3(-3)
3x + 3y = -9 (Equation 3)
Now add Equation 2 and Equation 3:
(2x - 3y) + (3x + 3y) = 4 + (-9)
2x - 3y + 3x + 3y = -5
5x = -5
x = -1
Now substitute the value of x back into Equation 1:
(-1) + y = -3
y = -3 + 1
y = -2
Therefore, the values of x and y are x = -1 and y = -2.