Which values of $x$ and $y$ would make the following expression represent a real number?
$(4+5)(x+y i)$
A. $x=4, y=5$
B. $x=-4, y=0$
C. $x=4, y=-5$
D. $x=0, y=5$
Answer (2)
Simplify the expression: ( 4 + 5 ) ( x + y i ) = 9 x + 9 y i .
Set the imaginary part to zero: 9 y = 0 .
Solve for y : y = 0 .
The solution is x = − 4 , y = 0 , which makes the expression a real number: x = − 4 , y = 0 .
Explanation
Understanding the Problem We are given the expression ( 4 + 5 ) ( x + y i ) and we want to find the values of x and y that make this expression a real number. A complex number is real if its imaginary part is zero.
Simplifying the Expression First, simplify the expression: ( 4 + 5 ) ( x + y i ) = 9 ( x + y i ) = 9 x + 9 y i
Setting the Imaginary Part to Zero For the expression 9 x + 9 y i to be a real number, the imaginary part must be zero. This means that 9 y must be equal to zero: 9 y = 0
Solving for y Solving for y , we get: y = 0 The value of x can be any real number since it only affects the real part of the expression.
Checking the Options Now, we check the given options to see which one satisfies the condition y = 0 :
x = 4 , y = 5 : ye q 0
x = − 4 , y = 0 : y = 0
x = 4 , y = − 5 : ye q 0
x = 0 , y = 5 : ye q 0
The only option that satisfies the condition y = 0 is x = − 4 , y = 0 .
Final Answer Therefore, the values of x and y that make the expression a real number are x = − 4 and y = 0 .
Examples
In electrical engineering, complex numbers are used to represent alternating current (AC) circuits. The impedance of a circuit, which is the opposition to the flow of current, is a complex number. If you want to design a circuit with a purely resistive impedance (i.e., no reactance), you need to ensure that the imaginary part of the impedance is zero. This problem demonstrates how to find values that make a complex expression real, which is crucial in such applications.
The values of x and y that make the expression ( 4 + 5 ) ( x + y i ) a real number are x = − 4 and y = 0 . Therefore, the correct multiple-choice option is B. This is because the imaginary part of the expression must be zero for it to be a real number.
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