Question

Solve for all possible values of the real numbers $x$ and $y$ in the following equations. $$x+iy=2i-7$$

Short answer

The solutions are $x=-7$ and $y=2$.

Step-by-step solution

Separate Real and Imaginary Parts

Given the equation: $$x+iy=2i-7$$ Separate the equation into its real and imaginary parts. This gives: $$x=-7$$ (Comparing real parts) $$y=2$$ (Comparing imaginary parts)

Check the Solutions

Verify the solutions found by substituting them back into the original equation: Let $x=-7$ and $y=2$, then: $$-7+i\times 2=-7+2i$$ The original equation is $$2i-7$$ Since $-7+2i=2i-7$, the solutions are correct.

Key concepts

Real Numbers

Real numbers are all the numbers we usually think of when dealing with everyday math. These include positive numbers, negative numbers, and zero. They are also called 'reals' to differentiate them from complex numbers. In the given equation, the real part is the number without the imaginary unit 'i'. For example, in the equation $x+iy=2i-7$, $x=-7$ is the real part because it does not involve the imaginary unit 'i'. Separating real numbers is crucial in solving equations that mix both real and imaginary numbers. This helps us isolate different parts of the equation and solve for each variable separately.

Imaginary Numbers

Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit 'i'. The imaginary unit is defined by the property that \i^2 = -1\. This means that \i\ is the square root of -1, a number that does not exist within the set of real numbers. In our equation $x+iy=2i-7$, the imaginary part is \y \times i\. By comparing the imaginary parts from both sides of the equation, we determined that \y = 2\. Imaginary numbers are essential in fields such as engineering and physics where they help in handling oscillations, waves, and other phenomena.

Complex Numbers

Complex numbers combine both real and imaginary parts into a single number of the form \a + bi\, where \a\ and \b\ are real numbers. In our exercise, $x+iy$ and \2i - 7\ are both complex numbers. To solve such an equation, we break it down into its real and imaginary components. By isolating the real and imaginary parts, we can deal with them separately. For example, in \x + iy = 2i - 7\, comparing the real parts gave us \x = -7\ and comparing the imaginary parts told us that \y = 2\. These techniques help us handle problems that would otherwise be difficult to solve using only real numbers.

Solution Verification

After finding the values of $x$ and $y$, it is crucial to verify the solution to ensure it satisfies the original equation. This process helps confirm that no mistakes were made during calculations. In our problem, we substituted $x=-7$ and $y=2$ back into the original equation: \x + i y = -7 + 2i\. This matches our original equation \2i - 7\, thus verifying our solution is correct. This step is important because it helps catch any errors and ensures the solution is reliable. Always make a habit of checking your work.