Which values of $x$ and $y$ would make the following expression represent a real number?
$(4+5i)(x+yi)$
A. $x=-4, y=0$
B. $x=4, y=-5$
C. $x=0, y=5$
Answer (2)
The values of x and y that make the expression ( 4 + 5 i ) ( x + y i ) a real number are x = 4 and y = − 5 . This is determined by setting the imaginary part of the expanded expression to zero. Therefore, the answer is option B.
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Simplify the expression: ( 4 + 5 ) ( x + y ) = 9 ( x + y ) .
Substitute x = − 4 and y = 0 : 9 ( − 4 + 0 ) = − 36 , which is a real number.
Substitute x = 4 and y = − 5 : 9 ( 4 − 5 ) = − 9 , which is a real number.
Substitute x = 0 and y = 5 : 9 ( 0 + 5 ) = 45 , which is a real number. Since all three pairs result in a real number, all three are valid solutions.
Explanation
Understanding the Problem We are given the expression ( 4 + 5 ) ( x + y ) and asked to find which of the given pairs of x and y will make the expression a real number. A real number is any number that can be found on the number line. This includes integers, rational numbers, and irrational numbers.
Simplifying the Expression First, simplify the expression: ( 4 + 5 ) ( x + y ) = 9 ( x + y ) Now, we will substitute each pair of x and y into the simplified expression and check if the result is a real number.
Checking the First Pair Case 1: x = − 4 and y = 0 9 ( x + y ) = 9 ( − 4 + 0 ) = 9 ( − 4 ) = − 36 Since − 36 is a real number, this pair is a valid solution.
Checking the Second Pair Case 2: x = 4 and y = − 5 9 ( x + y ) = 9 ( 4 + ( − 5 )) = 9 ( − 1 ) = − 9 Since − 9 is a real number, this pair is a valid solution.
Checking the Third Pair Case 3: x = 0 and y = 5 9 ( x + y ) = 9 ( 0 + 5 ) = 9 ( 5 ) = 45 Since 45 is a real number, this pair is a valid solution.
Conclusion Since all three pairs result in a real number, all three are valid solutions.
Examples
Understanding real numbers is crucial in many real-world applications. For instance, when calculating distances, areas, or volumes, the results are always real numbers. Similarly, in financial calculations, such as determining profits, losses, or interest rates, the outcomes are real numbers. Knowing how to work with real numbers ensures accurate and meaningful results in various practical scenarios.
