Cramer's rule

Cramer's rule is a way to help us solve math problems when we have a system of equations with multiple variables. It's named after a person named Gabriel Cramer who came up with it a long time ago.

Imagine you have two equations that look like this:

2x + 3y = 10
4x - 5y = 6

You might want to figure out what values of x and y would make both of these equations true at the same time. This is called the "solution" to the system of equations.

Cramer's rule works by using something called determinants. A determinant is like a special type of number that we can calculate for certain matrices (don't worry about what a matrix is for now).

For a system of two equations like the one above, we can use determinants to calculate the values of x and y that would make both equations true.

First we need to calculate a bunch of different determinants. For this system of equations, we need to calculate three determinants:

- The determinant of the "coefficient matrix." In this case, the coefficient matrix is:

2 3
4 -5

To calculate the determinant of this matrix, we multiply the top left and bottom right numbers (2 and -5) and then subtract the product of the top right and bottom left numbers (3 and 4).

2 * (-5) - 3 * 4 = -22

So the determinant of the coefficient matrix is -22.

- The determinant of the "x matrix." This is the same as the coefficient matrix, but with the column of x values replacing the x coefficients in the equation. So it looks like this:

10 3
6 -5

To calculate the determinant of this matrix, we use the same method as before:

10 * (-5) - 3 * 6 = -51

So the determinant of the x matrix is -51.

- The determinant of the "y matrix." This is the same as the coefficient matrix, but with the column of y values replacing the y coefficients in the equation. So it looks like this:

2 10
4 6

To calculate the determinant of this matrix:

2 * 6 - 10 * 4 = -28

So the determinant of the y matrix is -28.

Now we can use these determinants to find the values of x and y.

To find x, we take the determinant of the x matrix (-51) and divide it by the determinant of the coefficient matrix (-22):

-51 / -22 = 2.318

So x = 2.318.

To find y, we take the determinant of the y matrix (-28) and divide it by the determinant of the coefficient matrix (-22):

-28 / -22 = 1.273

So y = 1.273.

Therefore, the solution to this system of equations is x = 2.318 and y = 1.273.

Overall, Cramer's rule is a way to solve systems of equations by using determinants to find the values of each variable.