Macaulay's resultant

Macaulay's Resultant is like a magic trick that helps you solve two equations with two variables at the same time. Imagine you have two equations that look like this:

2x + 3y = 5
4x + 6y = 9

You could solve for x in the first equation and plug it into the second equation to get an answer, but that's a lot of work. Macaulay's Resultant makes it easier!

First, you take the coefficients of the variables in each equation and make them into something called a polynomial. For the first equation, you get 2x + 3y = 5, so the polynomial is 2x + 3y - 5. For the second equation, you get 4x + 6y = 9, so the polynomial is 4x + 6y - 9.

Then you write these two polynomials next to each other like this:
2x + 3y - 5 | 4x + 6y - 9

Now comes the magic part called Macaulay's Resultant. You take the first polynomial and repeat it once underneath like this:

2x + 3y - 5 | 4x + 6y - 9
2x + 3y - 5 | 4x + 6y - 9

But this time you shift the second polynomial to the right, so it lines up with the x term of the first polynomial:

2x + 3y - 5 | 4x + 6y - 9
4x + 6y - 9 |

Now you need to multiply the first column of the first polynomial by the second row of the second polynomial, and add it to the second column of the first polynomial times the first row of the second polynomial.

(2x)(6y-9) + (3y-5)(4x) = 12xy - 18x + 12xy - 20x - 15y + 25

This result is called the Macaulay's resultant. If the resultant is equal to zero, then there is a solution for the two equations, and the x and y values can be found by solving the polynomials.

That's it - you did it! You used Macaulay's Resultant to solve two equations with two variables at the same time.