Problem 5 - College Algebra and Calculus: An Applied Approach

Question

Find the real numbers \(a\) and \(b\) such that the equation is true. $$(a+3)+(b-1) i=7-4 i$$

Step-by-Step Solution

Answer

The solution to the equation is \(a = 4\) and \(b = -3\).

Step 1: Understand Complex Numbers and Identify Real and Imaginary Parts

Complex numbers are usually of the form \(z = a + bi\), where \(a\) and \(b\) denote the real and imaginary parts respectively. So, from the given equation, we find that the real parts are \(a+3\) and \(7\), and the imaginary parts are \(b-1\) and \(-4\).

Step 2: Equate and Solve for Real Parts

We equate the real part of both sides to find the value of \(a\). So we get: \(a + 3 = 7\). Solving for \(a\), we subtract \(3\) from both sides and get \(a = 7 - 3 = 4\).

Step 3: Equate and Solve for Imaginary Parts

Next, we equate the imaginary parts to find the value of \(b\). So we get: \(b - 1 = -4\). Solving for \(b\), we add \(1\) to both sides and get \(b = -4 + 1 = -3\).

Key Concepts

Real Numbers

Real numbers are a fundamental part of complex numbers. They can be whole numbers, fractions, or decimals that you'd find on a number line. In the context of our exercise, we're working with the real parts of a complex equation. When you see a complex number like \(z = a + bi\), the "\(a\)" is your real number. It's important to recognize that in complex equations, like the one given, the real part remains separate from the imaginary part.

• For our exercise, the real components are \(a+3\) and \(7\).

To solve this, we simply equate these real numbers. For example, to find \(a\), we set \(a+3 = 7\). Then, a little subtraction helps us find that \(a = 4\). It’s that simple! Breaking down the real numbers thoroughly allows us to solve this equation step by step.

Imaginary Parts

Imaginary parts of complex numbers bring an interesting twist to traditional math. Imaginary numbers are multiples of "i" where \(i^2 = -1\). In complex numbers like \(z = a + bi\), "\(bi\)" is your imaginary part. In our given problem, the imaginary part involves \(b-1\) and \(-4\). This part is just as crucial as the real part, because complex problems rely on both.

• For solving, the process includes setting the imaginary components equal: \(b - 1 = -4\).

Add \(1\) to each side to isolate \(b\), resulting in \(b = -3\). Imaginary numbers might be less tangible, but they are useful in many fields like engineering and physics.

Equations

Equations are mathematical statements that assert the equality of two expressions. They are the foundations upon which problem-solving in mathematics is built. In this exercise, we deal with an equation involving a complex number, where both sides of the equation contain real and imaginary parts. Each part must be solved separately to ensure accuracy.

• Start with equating the real parts to solve for a number.
• Then, equate the imaginary parts for the other number.

The resulting solutions, \(a = 4\) and \(b = -3\), satisfy both expressions of the equation. Working with equations of complex numbers requires understanding that both parts of the equation must be balanced, just as in simple algebraic equations.