Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Find the values of \(x\) and \(y\), where \(x\) and \(y\) are $$ 8+(3 x+y) i=2 x-4 i $$

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(y = -16\).

Step 1: Equate Real Parts

In the given equation, we have a complex number on both sides. For the real part of the equation, compare the real parts on each side of the equation: \[ 8 = 2x \].

Step 2: Solve for x

We now solve the equation for \(x\) obtained from the real parts: \[ 8 = 2x \]Divide both sides by 2:\[ x = 4 \].

Step 3: Equate Imaginary Parts

Next, compare the imaginary parts of both sides of the equation: \[ 3x + y = -4 \].

Step 4: Substitute x into Imaginary Part Equation

Substitute the value of \(x\) found in Step 2, which is 4, into the imaginary parts equation:\[ 3(4) + y = -4 \].

Step 5: Solve for y

Calculate the multiplication: \[ 12 + y = -4 \].Subtract 12 from both sides:\[ y = -16 \].

Key Concepts

Complex Numbers

Complex numbers are a fascinating area of mathematics that extend the idea of the one-dimensional number line to the two-dimensional complex plane. A complex number takes the form of \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, defined by the property \( i^2 = -1 \). Here, \( a \) is known as the real part and \( b \) is the imaginary part.
This number system is essential for various applications in fields such as engineering, physics, and computer science.

• Real Part: Represents the "traditional" number when considered alone.
• Imaginary Part: Scales the imaginary unit \( i \), creating new values.
• Complex Plane: A plane where the horizontal axis represents real numbers, while the vertical axis represents imaginary numbers.

Understanding how to handle these parts separately is essential when dealing with complex equations, as demonstrated in the provided exercise.

Equating Real and Imaginary Parts

One powerful method for solving complex equations is by equating their real and imaginary parts. When you have a complex equation, you often want to separate it into two simpler equations: one for the real parts and one for the imaginary parts.

This works because if two complex numbers are equal, their real parts must be equal and their imaginary parts must be equal separately.

• Real Equation: Formed by setting the real parts of the complex numbers equal.
• Imaginary Equation: Formed by equating the imaginary parts.

In the original problem, we see this method clearly during the steps of solving for \(x\) and \(y\). By separating the equation into \(8 = 2x\) and \(3x + y = -4\), we simplify the problem into two manageable parts, making the solution process much clearer.

Solving Linear Equations

Equations like \(8 = 2x\) involve finding the value of a variable that makes the equation true. These are known as linear equations because they can be graphically represented as straight lines on a coordinate plane.

The basic form is \(ax + b = c\), where you must isolate \(x\) to solve the equation. In our example, dividing both sides by 2 gives \(x = 4\).

Here are some general steps for solving linear equations:

• Move all terms involving the variable to one side.
• Place all constant terms on the other side.
• Simplify both sides of the equation.
• Divide or multiply to solve for the variable.

Applying these steps can quickly lead you to the solution, as we saw with \(x = 4\). Such equations form the basis of more complex algebraic problems.

Algebraic Manipulation

Algebraic manipulation forms the backbone of solving complex equations. It involves rearranging and simplifying expressions to make them easier to work with.

In our specific problem, we used algebraic manipulation by substituting \(x = 4\) into the equation \(3x + y = -4\). Changing \(3(4) + y = -4\) to \(12 + y = -4\) showcases how substitution and simplification help resolve variables.

• Substitution: Replacing a variable with a known value.
• Simplification: Combining like terms and reducing expressions.
• Re-arranging: Moving terms to isolate variables on one side of the equation.

These steps are crucial in breaking down more significant, seemingly complicated problems into smaller, more manageable parts. Mastery of these techniques is essential for success in algebra and beyond.