"i" representing the square root of negative 1

## Find real numbers x and y such that (2x+3y) + (3-5x)i = 1 -7i

# 1 Answer

We are asked to find x and y such that the problem statement is true. So we need two equations to be able to solve for the two unknowns. The trick in this problem is that we are dealing with complex numbers. Complex numbers have both a real part (without "i") and an imaginary part (with "i"). So to get our first equation we have to match the real part (the part without an "i") on the left side of the equation to the real part on the right hand side. Doing this gives

2x + 3y = 1 (1)

Now matching the imaginary parts (the part with an "i") on both sides of the equation gives our second equation:

3 - 5x = -7 (2)

Since the second equation only involves one variable, let's solve that first. Subtract 3 from both sides to get

-5x = -10

Then divide both sides by -5 to get

**x = 2**

Plugging our value of x into equation (1) gives

2(2) + 3y = 1

4 + 3y = 1

Subtract both sides by 4 to get

3y = -3

Finally divide both sides by 3 to get

**y = -1**