More About Complex Numbers:
Solve x and y
4x+5y+13i=7+(6x+5y)i
Please show work because I had a hard time finding the answer
To solve for x and y in the equation 4x + 5y + 13i = 7 + (6x + 5y)i, we need to separate the real and imaginary parts of the equation.
First, let's equate the real (non-imaginary) parts:
4x + 5y = 7 -- (Equation 1)
Next, let's equate the imaginary parts:
13i = (6x + 5y)i -- (Equation 2)
To eliminate the i, we can equate the coefficients of i on both sides of Equation 2:
13 = 6x + 5y -- (Equation 3)
Now, we have a system of two equations (Equations 1 and 3) that can be solved simultaneously.
To solve the system, we can use substitution or elimination method. Let's use elimination:
Multiply Equation 1 by 5:
20x + 25y = 35 -- (Equation 4)
Now, subtract Equation 3 from Equation 4:
20x + 25y - (6x + 5y) = 35 - 13
Combine like terms:
14x + 20y = 22 -- (Equation 5)
Now, we have a new equation (Equation 5) with only variables x and y.
We can isolate one variable (let's choose x) in terms of the other:
14x = 22 - 20y
Divide both sides by 14:
x = (22 - 20y) / 14 -- (Equation 6)
After isolating x, we can substitute this equation into one of the previous equations (Equation 1) to solve for y.
Substitute Equation 6 into Equation 1:
4((22 - 20y)/14) + 5y = 7
Simplify:
(88 - 80y)/14 + 5y = 7
Multiply both sides by 14:
88 - 80y + 70y = 98
Combine like terms:
-10y = 10
Divide both sides by -10:
y = -1
Now, substitute the value of y into Equation 6 to solve for x:
x = (22 - 20(-1))/14
Simplify:
x = (22 + 20)/14
x = 42/14
Simplify further:
x = 3
Therefore, the solution to the system of equations 4x + 5y + 13i = 7 + (6x + 5y)i is x = 3 and y = -1.