Question

. \((x+2 y+3)+i(3 x-y-1)=0\)

Short answer

x = -\frac{41}{7}, y = \frac{10}{7}

Step-by-step solution

Separate the Real and Imaginary Parts

Given the equation \( (x + 2y + 3) + i(3x - y - 1) = 0 \), separate it into its real and imaginary parts. This gives us two separate equations: \( x + 2y + 3 = 0 \) and \( 3x - y - 1 = 0 \)

Solve for x from the First Equation

From the equation \( x + 2y + 3 = 0 \), rearrange to solve for \( x \): \( x = -2y - 3 \)

Substitute x into the Second Equation

Substitute \( x = -2y - 3 \) into the second equation: \( 3(-2y - 3) - y - 1 = 0 \). Simplify and solve for \( y \)

Simplify and Solve for y

Distribute and combine like terms to simplify: \( -6y - 9 - y - 1 = 0 \), which simplifies to \( -7y - 10 = 0 \). Solve for \( y \): \( y = -\frac{10}{-7} = \frac{10}{7} \)

Solve for x using the value of y

Substitute \( y = \frac{10}{7} \) back into \( x = -2y - 3 \): \( x = -2 \times \frac{10}{7} - 3 \). Simplify to find \( x = -\frac{20}{7} - 3 = -\frac{20}{7} - \frac{21}{7} = -\frac{41}{7} \)

Key concepts

Separating Real and Imaginary Parts

To solve complex equations, it's important to separate the real and imaginary components. Complex numbers have a real part and an imaginary part. For example, in the equation eeds (x+2y+3)+i(3x-y-1)=0eeds, the expression before the 'i' is the real part, and the expression after the 'i' is the imaginary part. Remember, 'i' stands for the square root of -1. In order to solve the equation when it's equal to zero, both the real parts and the imaginary parts must independently equal zero.

This means you need to create two separate equations: one for the real parts and one for the imaginary parts. For our example:
Real part: (x+2y+3) = 0eeds
Imaginary part: (3x-y-1) = 0eeds

We now have two simpler linear equations to work with instead of one complex one.

Solving Linear Equations

Linear equations are equations of the first degree, meaning they have variables raised only to the power of one. Solving these equations typically involves isolating one of the variables. In our case, we have: (x + 2y + 3 = 0)eeds and (3x - y - 1 = 0)eeds.

The goal is to express one variable in terms of the other or find their specific values. By solving the first equation, we find that (x = -2y - 3)eeds. This expression for 'x' is then used in the second equation to simplify solving for 'y'.

Substitution Method

The substitution method is a technique for solving systems of equations where one equation is solved for one variable, and this expression is substituted into the other equation. From our problem, we already have (x = -2y - 3)eeds. Now, substitute eeds(-2y - 3)eeds for 'x' in the second equation:
eeds3(-2y - 3) - y - 1 = 0eeds.

After substitution, simplify and solve for 'y'. This involves distributing the constant through the parentheses and combining like terms. This gives us a single linear equation in one variable, making it easier to solve.

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They can be written in the form (a + bi)eeds, where 'a' is the real part and 'b' is the imaginary part. Complex numbers are used widely in engineering, physics, and applied mathematics. Understanding how to work with them involves separating their real and imaginary parts, solving linear equations, and using methods like substitution.

In summary, handling complex numbers often simplifies into dealing with two related linear equations. By mastering the techniques associated with separating the real and imaginary parts, solving linear equations, and using substitution, you can effectively solve various problems involving complex numbers.