Solve for x 8ix=-5
The problem given is an equation that needs to be solved for the variable x. The equation consists of a multiplication of a complex number, 8ix, and this expression is set equal to -5, which is a real number. The task is to perform algebraic manipulations to isolate x and find its value in terms of real and/or imaginary components.
$8 i x = - 5$
Answer
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:
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.
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.
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 $-8i$.
Complex Numbers: The imaginary unit $i$ is defined such that $i^2 = -1$. This property is used to simplify the equation further.
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.
