- Distribute the constant 22 and group the real and imaginary parts of the equation: \((22x + 28) + (22y + 4)i = 72 - 62i\).
- Separate the equation into two linear systems: \(22x + 28 = 72\) and \(22y + 4 = -62\).
- Solve the first equation for \(x\), which gives \(x = 2\).
- Solve the second equation for \(y\), which gives \(y = -3\).
- Final answer: \(\boxed{x=2, y=-3}\).
Explanation
- Analysis of the Problem
The problem asks to find the values of the real variables \(x\) and \(y\) in the complex equation \(22(x+yi) + (28+4i) = 72-62i\). A complex number is represented in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. To solve this, we will expand the equation, group the real and imaginary parts, and then set up a system of linear equations to solve for \(x\) and \(y\).
- Expansion and Setting Up Equations
First, distribute the constant 22 into the expression \(x+yi\):
\[22x + 22yi + 28 + 4i = 72 - 62i\]
Now, group the real terms and the imaginary terms on the left side:
\[(22x + 28) + (22y + 4)i = 72 - 62i\]
By comparing the real and imaginary parts of both sides, we get two separate linear equations:
For the real parts:
\[22x + 28 = 72\]
For the imaginary parts:
\[22y + 4 = -62\]
- Solving for Variables and Conclusion
Solve the real part equation for \(x\):
\[22x = 72 - 28\]
\[22x = 44\]
\[x = \frac{44}{22} = 2\]
Solve the imaginary part equation for \(y\):
\[22y = -62 - 4\]
\[22y = -66\]
\[y = \frac{-66}{22} = -3\]
Therefore, the values are \(x = 2\) and \(y = -3\). This matches the first option.
Examples
Complex numbers are vital in electrical engineering for analyzing alternating current (AC) circuits, where voltage and current are often represented as complex numbers to account for phase differences. For instance, solving equations similar to this one allows engineers to determine the impedance of a circuit component or the signal behavior in a system, ensuring devices operate safely and efficiently. By breaking down the real and imaginary components, engineers can simplify complex wave interactions into manageable linear equations.