Solving (18 + P) + I(90 - 3p) = 0 A Step-by-Step Guide
In the realm of complex numbers, equations often present a unique and intriguing challenge. One such challenge arises when we encounter an equation like (18 + p) + i(90 - 3p) = 0. This equation, while seemingly simple, involves the interplay of real and imaginary components. The task at hand is to solve for 'p', which means finding the value or values of 'p' that satisfy this equation. This exploration delves into the fundamental principles of complex numbers and equation solving, providing a step-by-step guide to unraveling the solution.
Understanding Complex Numbers
Before we dive into solving the equation, it's crucial to grasp the essence of complex numbers. A complex number is typically expressed in the form a + bi, where 'a' represents the real part and 'b' represents the imaginary part. The symbol 'i' signifies the imaginary unit, defined as the square root of -1 (√-1). Complex numbers extend the realm of real numbers by incorporating this imaginary component. This allows us to work with solutions to equations that have no real number solutions, such as the square root of negative numbers.
The Equation (18 + p) + i(90 - 3p) = 0
Our focus is on the equation (18 + p) + i(90 - 3p) = 0. This equation states that a complex number, composed of a real part (18 + p) and an imaginary part (90 - 3p) multiplied by i, equals zero. For a complex number to be zero, both its real and imaginary parts must independently be equal to zero. This principle forms the bedrock of our solution strategy.
Setting Up the Equations
To solve for 'p', we leverage the principle that both the real and imaginary parts of the complex number must be zero. This leads us to two separate equations:
- Real Part: 18 + p = 0
- Imaginary Part: 90 - 3p = 0
We now have a system of two linear equations with one unknown variable, 'p'. Solving this system will provide the value(s) of 'p' that satisfy the original complex equation.
Solving the Real Part Equation
The first equation, 18 + p = 0, is a straightforward linear equation. To isolate 'p', we subtract 18 from both sides of the equation:
p = -18
This yields a potential solution for 'p'. However, it's crucial to verify if this value also satisfies the imaginary part equation.
Solving the Imaginary Part Equation
The second equation, 90 - 3p = 0, also represents a linear relationship. To solve for 'p', we first add 3p to both sides:
90 = 3p
Next, we divide both sides by 3:
p = 30
This gives us another potential solution for 'p'. Now, we have two distinct values for 'p' obtained from the real and imaginary parts of the equation.
Verifying the Solutions
We have found two potential solutions for 'p': p = -18 from the real part equation and p = 30 from the imaginary part equation. However, for 'p' to be a valid solution to the original complex equation, it must simultaneously satisfy both the real and imaginary part equations. This means that the value of 'p' must make both the real and imaginary parts of the complex number equal to zero.
Let's substitute p = -18 back into the imaginary part equation:
90 - 3(-18) = 90 + 54 = 144
Since 144 is not equal to zero, p = -18 does not satisfy the imaginary part equation and is therefore not a valid solution.
Now, let's substitute p = 30 back into the real part equation:
18 + 30 = 48
Since 48 is not equal to zero, p = 30 does not satisfy the real part equation and is also not a valid solution.
The Critical Insight: Simultaneous Satisfaction
The verification step reveals a crucial insight: for the complex equation to hold true, both the real and imaginary parts must be zero simultaneously. We initially treated the real and imaginary parts as separate equations, solving for 'p' in each independently. However, the original equation demands a single value of 'p' that makes both parts vanish. Since we found different values of 'p' for each part, and neither value satisfies the other part's equation, it implies that there is no single value of 'p' that can make the entire complex expression equal to zero.
The Correct Approach: A System of Equations
The realization that 'p' must satisfy both equations simultaneously points us towards the correct approach: we are dealing with a system of equations. To reiterate, our system is:
- 18 + p = 0
- 90 - 3p = 0
We solved these equations individually, but to find a true solution for the complex equation, we need a value of 'p' that works for both equations. Let's re-examine our solutions and the implications.
Reassessing the Solutions
We found p = -18 from the first equation and p = 30 from the second equation. The fact that these values are different indicates that the system of equations is inconsistent. An inconsistent system is one that has no solution; there is no value of 'p' that can satisfy both equations at the same time.
Graphical Interpretation (Optional but Illuminating)
To further understand this, we can visualize these equations graphically. If we were to plot these as lines on a graph (considering them as functions where the result is zero), we would see that the lines intersect at no point. The first equation, 18 + p = 0, can be seen as a vertical line at p = -18. The second equation, 90 - 3p = 0, can be rearranged to p = 30, which is another vertical line. Since parallel lines never intersect, there is no common solution.
The Final Answer: No Solution
Given our analysis, we conclude that there is no value of 'p' that satisfies the equation (18 + p) + i(90 - 3p) = 0. The real and imaginary parts of the equation lead to conflicting requirements for 'p', making it impossible to find a solution. This conclusion highlights the importance of verifying solutions in the context of complex numbers, where the interplay between real and imaginary components dictates the validity of a solution.
Solving for 'p' in the complex equation (18 + p) + i(90 - 3p) = 0 has been a journey through the intricacies of complex numbers and equation solving. We began by understanding the nature of complex numbers and the condition for a complex number to be zero. We then translated the complex equation into a system of two linear equations, one for the real part and one for the imaginary part. Solving these equations individually led to conflicting values for 'p', revealing that the system was inconsistent. Ultimately, we concluded that there is no solution for 'p' that satisfies the original complex equation. This exploration underscores the critical principle that solutions in complex number equations must simultaneously satisfy both the real and imaginary components, a principle that governs the behavior of these fascinating mathematical entities.
