Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 35-38: Find the values of \(x\) and \(y\), where \(x\) and \(y\) are real numbers. $$ (x-y)+3 i=7+y i $$

Step-by-Step Solution

Answer

\( x = 10 \), \( y = 3 \).

Step 1: Equate the Real Parts

In the given equation, the real part on the left is \( x-y \) and on the right is 7. This gives the equation: \( x - y = 7 \).

Step 2: Equate the Imaginary Parts

In the given equation, the imaginary part on the left is 3, and on the right is \( y \). Thus, we can set up the equation: \( 3 = y \).

Step 3: Solve for y

From Step 2, we already have \( y = 3 \). This value is found directly by equating the imaginary parts.

Step 4: Substitute y and Solve for x

Use the value of \( y \) from Step 3 in the equation from Step 1: \( x - 3 = 7 \). Solve for \( x \) by adding 3 to both sides, giving \( x = 10 \).

Key Concepts

Real Numbers

Real numbers are those that we commonly use in everyday life. They include

• natural numbers (1, 2, 3, ...),
• whole numbers (0, 1, 2, 3, ...),
• integers (..., -2, -1, 0, 1, 2, ...),
• rational numbers (fractions like 1/2, 2/3),
• and irrational numbers (√2, π).

These numbers are found on the number line and do not include any imaginary components.
When solving algebraic equations that involve complex numbers like \((x-y)+3i=7+yi\), you'll often need to separately consider the real and imaginary parts. Here, the real components are expressed through the equation \(x-y = 7\). This means we are only focusing on the part of the equation without the 'i', the imaginary unit.

Imaginary Numbers

Imaginary numbers arise when we need to take square roots of negative numbers, which aren't possible with only real numbers. The imaginary unit, denoted as 'i', is the square root of -1.
An imaginary number is typically expressed as a real number multiplied by 'i', such as \(3i\). When equating the imaginary parts of the equation \((x-y)+3i=7+yi\), we look at the components including 'i'.
This helps us find the unknowns in our equation like identifying the value of \(y\), by setting\(3 = y\). Consider imaginary numbers as essential tools for solving equations that don't fit into the real number framework.

Algebraic Equations

Algebraic equations are mathematical statements showing the equality between two expressions. They can include numbers, variables, and operations. In the context of complex numbers, we break them down into their real and imaginary parts.
For example, the equation \((x-y)+3i = 7+yi\) can be split into two separate equations: one for real numbers \(x-y = 7\) and another for imaginary numbers \(3 = y\). Solving these equations involves finding the values of the unknowns that satisfy both portions.
This means we substitute to find each variable individually.

• Solve \(y\) from the imaginary part, which gives us \(y = 3\).
• Substitute \(y = 3\) into the real part equation \(x-y = 7\) to find \(x = 10\).

Understanding the components of algebraic equations allows you to break down complex problems into manageable parts for easier solving.