solve the following equations for real X and Y
1) (2-3i) (x+yi) = 4+i
2) (2-2i) (x+yi) = 2(x=2yi) + 2i - 1
need help :( after getting answer i m gonna write this mathematics in my death note lol
6 views!! but still no one answer ...-_-!
impatient much? whining after 10 minutes?
Have you been working on it while waiting for someone to help?
Two complex numbers are equal if their real and imaginary parts are equal
(2-3i) (x+yi) = 4+i
2x - 3xi + 2yi + 3y = 4+i
(2x+3y) + (-3x+2y) = 4+i
That means that
2x+3y = 4
-3x+2y = 1
x = 5/13 and y = 14/13
check: (2-3i)(5+14i)/13 = 4+i yes
assuming a typo, I used
(2-2i) (x+yi) = 2(x+2yi) + 2i - 1
(2x+2y) + (-2x+2y)i = (2x-1) + (4y+2)i
so now just solve
2x+2y = 2x-1
-2x+2y = 4y+2
No, that's what I was doing
i was trying to solve this and i got right answer
by the way thank you for your help :)
by the way are you teacher?
Sure, let me guide you through the process of solving these equations.
1) (2-3i)(x+yi) = 4+i
To solve this equation, we need to perform complex number multiplication. Let's expand the left side of the equation:
2x + 2yi - 3ix - 3y = 4 + i
Now, we can separate the real and imaginary parts of the equation:
(2x - 3y) + (2y - 3x)i = 4 + i
Equating the real and imaginary parts separately, we get two separate equations:
2x - 3y = 4 --(1)
2y - 3x = 1 --(2)
Now, we have a system of linear equations in x and y.
To solve this system, we can use any method such as substitution or elimination. Let's use the elimination method:
Multiply equation (1) by 2: 4x - 6y = 8 --(3)
Now, subtract equation (3) from equation (2):
2y - 3x - (4x - 6y) = 1 - 8
2y - 3x - 4x + 6y = -7
8y - 7x = -7 --(4)
Now, we have a new linear equation in x and y.
Let's rearrange equation (4) to solve for y in terms of x:
8y = 7x - 7
y = (7x - 7)/8
Now, substitute this expression for y in equation (1):
2x - 3[(7x - 7)/8] = 4
Simplify and solve for x:
16x - 21x + 21 = 32
-5x = 11
x = -11/5
Now, substitute this value of x into the equation we found for y:
y = (7(-11/5) - 7)/8
Simplify to find the value of y:
y = -5/4
Therefore, the solution to the equation is x = -11/5 and y = -5/4.
2) (2-2i)(x+yi) = 2(x+2yi) + 2i - 1
Let's expand the left side of the equation:
2x + 2yi - 2ix - 2y = 2x + 4yi + 2i - 1
Now, separate the real and imaginary parts:
(2x - 2y) + (2y - 2x)i = 2x + 4yi + 2i - 1
Equating real and imaginary parts separately, we get:
2x - 2y = -1 --(5)
2y - 2x = 4 --(6)
Again, we have a system of linear equations in x and y.
Using the elimination method, let's subtract equation (5) from equation (6):
2y - 2x - (2x - 2y) = 4 - (-1)
2y - 2x - 2x + 2y = 4 + 1
4y - 4x = 5 --(7)
Now, rearrange equation (7) to solve for y in terms of x:
4y = 4x + 5
y = (4x + 5)/4
Substitute this expression for y into equation (5):
2x - 2[(4x + 5)/4] = -1
Simplify and solve for x:
8x - 10x - 10 = -4
-2x = 6
x = -6/2
x = -3
Now, substitute this value of x into the equation we found for y:
y = (4(-3) + 5)/4
Simplify to find the value of y:
y = 7/4
Therefore, the solution to the equation is x = -3 and y = 7/4.
I hope this helps you solve the equations. Remember to double-check your solutions and let me know if you need any further assistance! And as for the death note reference, remember to use your mathematical powers for good!