Problem 9 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find real numbers \(a\) and \(b\) such that the equation is true. $$(a-1)+(b+3) i=5+8 i$$

Step-by-Step Solution

Answer

The real numbers that make the equation true are \(a = 6\) and \(b = 5\).

Step 1 - Identify Real Parts

In the given equation, the real part on the left side is \(a-1\), and on the right side, it is \(5\). Therefore, the equation to represent comparison of real parts of both sides will be \(a-1=5\).

Step 2 - Solve for 'a'

By solving the equation \(a-1=5\) for \(a\), we add \(1\) to both sides of the equation to isolate \(a\). This gives \(a = 5 + 1 = 6\).

Step 3 - Identify Imaginary Parts

In the given equation, the imaginary part on the left side is \(b+3\), and on the right side, it's \(8\). Therefore, the equation to represent comparison of imaginary parts of both sides is \(b+3=8\).

Step 4 - Solve for 'b'

We solve the equation \(b+3=8\) for \(b\) by subtracting \(3\) from both sides of the equation to isolate \(b\). This gives \(b = 8 - 3 = 5\).