Solve for x 8ix=-5

Solution:

Step 1:

Isolate $x$ by dividing the equation $8ix=-5$ by $8i$.

$$\frac{8ix}{8i}=\frac{-5}{8i}$$

Step 2:

Simplify the equation by reducing similar terms.

Step 2.1:

Eliminate the common factor of $8$.

$$\frac{\cancel{8}ix}{\cancel{8}i}=\frac{-5}{8i}$$

Step 2.2:

Remove the common factor of $i$.

$$\frac{\cancel{i}x}{\cancel{i}}=\frac{-5}{8i}$$ Thus, we have $x=\frac{-5}{8i}$.

Step 3:

Rationalize the denominator of the right-hand side.

Step 3.1:

Multiply both the numerator and denominator by the conjugate of $8i$.

$$x=\frac{-5}{8i}\cdot \frac{i}{i}$$

Step 3.2:

Perform the multiplication.

Step 3.2.1:

Combine terms.

$$x=\frac{-5i}{8ii}$$

Step 3.2.2:

Simplify the denominator.

Step 3.2.2.1:

Enclose the denominator in parentheses.

$$x=\frac{-5i}{8(ii)}$$

Step 3.2.2.2:

Apply the exponent rule.

$$x=\frac{-5i}{8i^2}$$

Step 3.2.2.3:

Recognize that $i^2=-1$.

$$x=\frac{-5i}{8\cdot -1}$$

Step 3.3:

Simplify the denominator.

$$x=\frac{-5i}{-8}$$

Step 3.4:

A negative divided by a negative is a positive.

$$x=\frac{5i}{8}$$

Knowledge Notes:

To solve the equation $8ix=-5$ for $x$, we follow a systematic approach:

1. Division to Isolate Variable: We start by isolating the variable $x$ by dividing both sides of the equation by $8i$. This is a basic algebraic technique to get the variable on one side of the equation.

2. Simplification: We simplify the equation by canceling out common factors. In this case, both $8$ and $i$ are common factors on the left side of the equation, and they can be canceled out.

3. Rationalizing the Denominator: Since we have a complex number in the denominator, we multiply the numerator and denominator by the complex conjugate of the denominator to make it a real number. The complex conjugate of $8i$ is $-8i$.

4. Complex Numbers: The imaginary unit $i$ is defined such that $i^2=-1$. This property is used to simplify the equation further.

5. Multiplication and Division of Negative Numbers: When we divide or multiply two negative numbers, the result is positive. This is a fundamental rule in arithmetic.

By following these steps, we can find the solution to the equation in a structured and logical manner.