Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
Sure, let's solve the given equation step by step. We're given:
[tex]\[ \frac{(1+i)x - 2i}{3+i} + \frac{(2 - 3i)y + i}{3-i} = i \][/tex]
The first step is to simplify each fraction. Let's simplify the denominator by multiplying the numerator and the denominator by the conjugate of the denominator.
### Simplifying the first term [tex]\(\frac{(1+i)x - 2i}{3+i}\)[/tex]:
Multiply the numerator and the denominator by the conjugate of [tex]\(3+i\)[/tex], which is [tex]\(3-i\)[/tex]:
[tex]\[ \frac{[(1+i)x - 2i](3-i)}{(3+i)(3-i)} \][/tex]
Since [tex]\( (3+i)(3-i) = 3^2 - i^2 = 9 - (-1) = 10 \)[/tex], we get:
[tex]\[ \frac{[(1+i)x - 2i](3-i)}{10} \][/tex]
Now expand the numerator:
[tex]\[ (1+i)x(3-i) - 2i(3-i) \][/tex]
Distribute [tex]\( (1+i)x \)[/tex]:
[tex]\[ = (3x - ix + 3ix - i^2 x) - 2i(3-i) \][/tex]
[tex]\[ = (3x + 2ix + x) - (6i + 2) \][/tex]
[tex]\[ = 4x + 2ix - 6i - 2 \][/tex]
Thus, the first term simplifies to:
[tex]\[ \frac{4x + 2ix - 6i - 2}{10} \][/tex]
This can be split into real and imaginary parts:
[tex]\[ \frac{4x - 2}{10} + \frac{2ix - 6i}{10} = \frac{4x - 2}{10} + \frac{2x - 6}{10}i \][/tex]
### Simplifying the second term [tex]\(\frac{(2 - 3i)y + i}{3-i}\)[/tex]:
Multiply the numerator and the denominator by the conjugate of [tex]\(3-i\)[/tex], which is [tex]\(3+i\)[/tex]:
[tex]\[ \frac{[(2 - 3i)y + i](3+i)}{(3-i)(3+i)} \][/tex]
Since [tex]\( (3-i)(3+i) = 3^2 - (-1) = 9 + 1 = 10 \)[/tex], we get:
[tex]\[ \frac{[(2 - 3i)y + i](3+i)}{10} \][/tex]
Now expand the numerator:
[tex]\[ (2 - 3i)y(3+i) + i(3+i) \][/tex]
Distribute [tex]\( (2 - 3i)y \)[/tex]:
[tex]\[ = (6y + 2iy - 9iy - 3i^2 y) + (3i + i^2) \][/tex]
[tex]\[ = (6y - 7iy + 3y) + (3i - 1) \][/tex]
[tex]\[ = 9y - 7iy + 3i - 1 \][/tex]
Thus, the second term simplifies to:
[tex]\[ \frac{9y - 1}{10} + \frac{-7iy + 3i}{10} \][/tex]
This can be split into real and imaginary parts:
[tex]\[ \frac{9y - 1}{10} + \frac{-7iy + 3i}{10} = \frac{9y - 1}{10} + \frac{-7y + 3}{10}i \][/tex]
### Combine and equate to [tex]\(i\)[/tex]:
Combining the simplified fractions:
[tex]\[ \left(\frac{4x - 2}{10} + \frac{9y - 1}{10}\right) + \left(\frac{2x - 6}{10} + \frac{-7y + 3}{10}\right)i = i \][/tex]
Equating the real parts and the imaginary parts separately:
Real part:
[tex]\[ \frac{4x - 2}{10} + \frac{9y - 1}{10} = 0 \][/tex]
Simplifies to:
[tex]\[ 4x - 2 + 9y - 1 = 0 \][/tex]
[tex]\[ 4x + 9y - 3 = 0 \quad \text{(1)} \][/tex]
Imaginary part:
[tex]\[ \frac{2x - 6}{10} + \frac{-7y + 3}{10} = 1 \][/tex]
Simplifies to:
[tex]\[ 2x - 6 - 7y + 3 = 10 \][/tex]
[tex]\[ 2x - 7y - 3 = 10 \][/tex]
[tex]\[ 2x - 7y = 13 \quad \text{(2)} \][/tex]
### Solve the system of equations:
We have the system of linear equations:
1. [tex]\( 4x + 9y = 3 \)[/tex]
2. [tex]\( 2x - 7y = 13 \)[/tex]
Multiply the second equation by 2:
[tex]\[ 4x - 14y = 26 \][/tex]
Subtract from the first equation:
[tex]\[ 4x + 9y - (4x - 14y) = 3 - 26 \][/tex]
[tex]\[ 4x + 9y - 4x + 14y = -23 \][/tex]
[tex]\[ 23y = -23 \][/tex]
[tex]\[ y = -1 \][/tex]
Substitute [tex]\(y = -1\)[/tex] into the first equation:
[tex]\[ 4x + 9(-1) = 3 \][/tex]
[tex]\[ 4x - 9 = 3 \][/tex]
[tex]\[ 4x = 12 \][/tex]
[tex]\[ x = 3 \][/tex]
Thus, the real values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are:
[tex]\[ \boxed{x = 3, y = -1} \][/tex]
[tex]\[ \frac{(1+i)x - 2i}{3+i} + \frac{(2 - 3i)y + i}{3-i} = i \][/tex]
The first step is to simplify each fraction. Let's simplify the denominator by multiplying the numerator and the denominator by the conjugate of the denominator.
### Simplifying the first term [tex]\(\frac{(1+i)x - 2i}{3+i}\)[/tex]:
Multiply the numerator and the denominator by the conjugate of [tex]\(3+i\)[/tex], which is [tex]\(3-i\)[/tex]:
[tex]\[ \frac{[(1+i)x - 2i](3-i)}{(3+i)(3-i)} \][/tex]
Since [tex]\( (3+i)(3-i) = 3^2 - i^2 = 9 - (-1) = 10 \)[/tex], we get:
[tex]\[ \frac{[(1+i)x - 2i](3-i)}{10} \][/tex]
Now expand the numerator:
[tex]\[ (1+i)x(3-i) - 2i(3-i) \][/tex]
Distribute [tex]\( (1+i)x \)[/tex]:
[tex]\[ = (3x - ix + 3ix - i^2 x) - 2i(3-i) \][/tex]
[tex]\[ = (3x + 2ix + x) - (6i + 2) \][/tex]
[tex]\[ = 4x + 2ix - 6i - 2 \][/tex]
Thus, the first term simplifies to:
[tex]\[ \frac{4x + 2ix - 6i - 2}{10} \][/tex]
This can be split into real and imaginary parts:
[tex]\[ \frac{4x - 2}{10} + \frac{2ix - 6i}{10} = \frac{4x - 2}{10} + \frac{2x - 6}{10}i \][/tex]
### Simplifying the second term [tex]\(\frac{(2 - 3i)y + i}{3-i}\)[/tex]:
Multiply the numerator and the denominator by the conjugate of [tex]\(3-i\)[/tex], which is [tex]\(3+i\)[/tex]:
[tex]\[ \frac{[(2 - 3i)y + i](3+i)}{(3-i)(3+i)} \][/tex]
Since [tex]\( (3-i)(3+i) = 3^2 - (-1) = 9 + 1 = 10 \)[/tex], we get:
[tex]\[ \frac{[(2 - 3i)y + i](3+i)}{10} \][/tex]
Now expand the numerator:
[tex]\[ (2 - 3i)y(3+i) + i(3+i) \][/tex]
Distribute [tex]\( (2 - 3i)y \)[/tex]:
[tex]\[ = (6y + 2iy - 9iy - 3i^2 y) + (3i + i^2) \][/tex]
[tex]\[ = (6y - 7iy + 3y) + (3i - 1) \][/tex]
[tex]\[ = 9y - 7iy + 3i - 1 \][/tex]
Thus, the second term simplifies to:
[tex]\[ \frac{9y - 1}{10} + \frac{-7iy + 3i}{10} \][/tex]
This can be split into real and imaginary parts:
[tex]\[ \frac{9y - 1}{10} + \frac{-7iy + 3i}{10} = \frac{9y - 1}{10} + \frac{-7y + 3}{10}i \][/tex]
### Combine and equate to [tex]\(i\)[/tex]:
Combining the simplified fractions:
[tex]\[ \left(\frac{4x - 2}{10} + \frac{9y - 1}{10}\right) + \left(\frac{2x - 6}{10} + \frac{-7y + 3}{10}\right)i = i \][/tex]
Equating the real parts and the imaginary parts separately:
Real part:
[tex]\[ \frac{4x - 2}{10} + \frac{9y - 1}{10} = 0 \][/tex]
Simplifies to:
[tex]\[ 4x - 2 + 9y - 1 = 0 \][/tex]
[tex]\[ 4x + 9y - 3 = 0 \quad \text{(1)} \][/tex]
Imaginary part:
[tex]\[ \frac{2x - 6}{10} + \frac{-7y + 3}{10} = 1 \][/tex]
Simplifies to:
[tex]\[ 2x - 6 - 7y + 3 = 10 \][/tex]
[tex]\[ 2x - 7y - 3 = 10 \][/tex]
[tex]\[ 2x - 7y = 13 \quad \text{(2)} \][/tex]
### Solve the system of equations:
We have the system of linear equations:
1. [tex]\( 4x + 9y = 3 \)[/tex]
2. [tex]\( 2x - 7y = 13 \)[/tex]
Multiply the second equation by 2:
[tex]\[ 4x - 14y = 26 \][/tex]
Subtract from the first equation:
[tex]\[ 4x + 9y - (4x - 14y) = 3 - 26 \][/tex]
[tex]\[ 4x + 9y - 4x + 14y = -23 \][/tex]
[tex]\[ 23y = -23 \][/tex]
[tex]\[ y = -1 \][/tex]
Substitute [tex]\(y = -1\)[/tex] into the first equation:
[tex]\[ 4x + 9(-1) = 3 \][/tex]
[tex]\[ 4x - 9 = 3 \][/tex]
[tex]\[ 4x = 12 \][/tex]
[tex]\[ x = 3 \][/tex]
Thus, the real values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are:
[tex]\[ \boxed{x = 3, y = -1} \][/tex]
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
